Uncover Excluded Values: A Guide To Ensuring Function Continuity

To find excluded values, examine the function for expressions involving division by zero or square roots of negative numbers. Solve these expressions for potential excluded values and exclude values outside the domain of the function. This process ensures that the function is defined and continuous over its domain, avoiding undefined or infinite results. Excluded values often lead to vertical asymptotes or holes in function graphs and are important for understanding the behavior and properties of the function.

Exploring the Intriguing World of Excluded Values

In the realm of mathematics, certain values play a unique role as excluded values. These are values that, when plugged into a function, result in undefined or illogical outcomes. Understanding these values is crucial for ensuring accuracy and precision in mathematical calculations.

Unraveling the Mystery

Excluded values arise from expressions that involve mathematical operations that are not feasible for certain inputs. The most common culprits are:

• Division by zero: When the denominator of a fraction is zero, division becomes impossible, leading to an undefined result. For instance, 5/0 has no meaningful value.
• Square roots of negative numbers: Negative numbers do not have real square roots within the realm of real numbers. This means that expressions like √(-4) are undefined.
• Logarithms of negative numbers: Logarithms are defined only for positive numbers. Attempting to find the log of a negative number, such as log(-2), yields an excluded value.

The Impact of Excluded Values

Excluded values have a significant impact on the behavior of functions. They can:

• Create vertical asymptotes: When a function involves division by an expression that equals zero at a particular value, a vertical asymptote occurs at that value. This is because the function’s output approaches infinity or negative infinity as the input gets close to the excluded value.
• Result in holes: Holes occur when a function’s graph has a point that is not defined but can be filled in with a different value. This happens when the function has an excluded value that is part of the function’s domain.

Uncovering Excluded Values

Finding excluded values is a crucial step in analyzing functions. The process involves:

1. Examining the Function: Identify any expressions that involve division by zero or square roots of negative numbers.
2. Setting Expressions Equal to Zero: Solve the identified expressions for potential excluded values.
3. Checking the Domain: Exclude any values outside the domain of the function.

A Practical Example

To illustrate, let’s consider the function f(x) = (x-3)/(x-2).

• Step 1: The denominator (x-2) cannot be zero, so we have an excluded value at x=2.
• Step 2: Setting x-2 = 0 gives x=2.
• Step 3: The domain of the function is all real numbers except x=2, so we exclude x=2.

Therefore, the function f(x) has an excluded value of x=2.

Excluded Values: Unlocking Mathematical Mysteries

In the realm of mathematics, there exist certain values that play a crucial role in shaping the behavior of functions and equations. These special values, known as excluded values, are like forbidden zones that cannot be part of the input or output of a function.

What Lurks in the Shadows of Excluded Values?

Division by Zero: Perhaps the most notorious excluded value is zero under the division operation. Think of it this way: if you try to divide any number by zero, you’re attempting to determine how many times zero goes into that number. But hold up! Zero can’t fit into any other number, much like trying to fit a square peg into a round hole.

Square Roots of Negative Numbers: The realm of complex numbers introduces another type of excluded value: the square roots of negative numbers. Unlike real numbers, which have two square roots, negative numbers don’t have any real square roots. It’s like trying to find the length of a shadow projected by a mirror – it simply doesn’t exist within the realm of real numbers.

Logarithms of Negative Numbers: Another forbidden territory is the realm of logarithms of negative numbers. Logarithms, which are essentially the inverse of exponents, only work for positive numbers. Trying to calculate the logarithm of a negative number is akin to trying to unlock a door with a non-existent key.

The Power of Excluded Values

Asymptotes: Excluded values often reveal themselves through their presence as vertical asymptotes on the graph of a function. Just like a fence that blocks your path, a vertical asymptote indicates a point where the function shoots off to infinity, rendering that excluded value unapproachable.

Holes: Sometimes, excluded values manifest as holes in the graph of a function. These holes represent missing points where a function is undefined. Think of a torn piece of paper – there’s a gap where the data should be, corresponding to the excluded value.

Unmasking Excluded Values: A Detective’s Guide

1. Examine the Function: Search for any expressions involving division by zero or square roots of negative numbers. These are potential culprits for excluded values.

2. Set Expressions Equal to Zero: Solve the expressions you identified in step 1 for potential excluded values.

3. Check the Domain: Determine the range of acceptable input values for the function. Excluded values will lie outside this domain.

Case Study: The Journey of Function f(x)

Let’s embark on a detective adventure to find the excluded values of the function f(x) = (x-2)/(x^2-4).

Division by Zero: We can’t divide by (x^2-4) if it equals zero. So, we solve x^2-4 = 0 and find x = 2, -2.

Check the Domain: The domain of f(x) is all real numbers except x = 2 and x = -2.

Hence, the excluded values of f(x) are x = 2 and x = -2.

Excluded Values: When Math Goes Off Limits

Division by Zero: A Mathematical Pitfall

When it comes to division, one rule stands out: never divide by zero. This seemingly simple concept holds profound implications in the world of mathematics. Division by zero, often denoted as 0/0, is an operation that has no meaningful result. It’s like trying to walk without taking any steps or asking for something without offering anything in return. The concept simply doesn’t add up.

Mathematically speaking, division by zero results in an undefined expression. Why is this so? Imagine trying to divide a pizza equally among zero people. How many slices would each person get? It’s an impossible scenario, right? The same principle applies to dividing any number by zero. There’s no way to determine a meaningful result.

The Consequences of Division by Zero

This mathematical quirk has important consequences. In many equations and functions, division by zero can create what’s known as an excluded value. An excluded value is a point where a function or expression is undefined. These values break the rules of mathematics and create a mathematical no-go zone.

For example, consider the equation: y = 1/x. This equation works fine for most values of x, but when x is equal to zero, we run into a problem. Dividing 1 by 0 results in an undefined expression, so x = 0 becomes an excluded value for this equation.

Finding Excluded Values

Identifying excluded values is crucial when working with mathematical equations and functions. To do this, we need to examine the function and identify any expressions that involve division by zero or other operations that can lead to undefined values.

Step 1: Examine the Function

Look for expressions that involve division by zero or square roots of negative numbers. These are potential sources of excluded values.

Step 2: Set Expressions Equal to Zero

Solve these expressions for potential excluded values. For example, if we have the equation y = 1/x, we would set the denominator x equal to zero to find the excluded value.

Step 3: Check the Domain

Finally, consider the domain of the function. The domain is the set of all possible values for the input variable. Exclude any values from the domain that make the function undefined.

Understanding excluded values is essential for navigating the world of mathematics. By recognizing these mathematical no-go zones, we can avoid pitfalls and ensure that our calculations and equations stand on solid mathematical ground.

Understanding Excluded Values: A Guide to Mathematical Exceptions

What are Excluded Values?

In the realm of mathematics, certain values are deemed “excluded” for particular functions. These excluded values are numbers that cannot be plugged into the function without causing the result to be undefined or produce an invalid calculation.

Causes of Excluded Values

Excluded values arise when a function involves certain operations that are not mathematically permissible for certain inputs. Here are some common causes:

• Division by zero: Dividing any number by zero results in an undefined expression.
• Square roots of negative numbers: Real numbers do not have real-valued square roots of negative numbers.
• Logarithms of negative numbers: The logarithm of a negative number is undefined.

Related Concepts

Excluded values have significant implications in mathematical analysis and graph theory:

• Asymptotes: Excluded values can create vertical asymptotes in the graph of a function, indicating that the function approaches infinity as the input approaches the excluded value.
• Holes: Excluded values can also lead to holes in the graph of a function, where the function is undefined at those values but is defined at other points.

Finding Excluded Values

To identify excluded values in a given function, follow these steps:

1. Examine the Function: Look for expressions involving division by zero or square roots of negative numbers.
2. Set Expressions Equal to Zero: Solve these expressions for potential excluded values.
3. Check the Domain: Exclude any values that are outside the domain of the function.

Example

Consider the function (f(x) = \frac{x-2}{x^2 – 4}).

• Step 1: Examining the function, we identify the term (\frac{x-2}{x^2 – 4}), which involves division by (x^2 – 4).
• Step 2: To find potential excluded values, we set (x^2 – 4 = 0). Solving for (x), we get (x = \pm 2).
• Step 3: The domain of the function is all real numbers except (x = \pm 2). Therefore, the excluded values are (x = \pm 2).

Unveiling the Secrets of Excluded Values

From the tantalizing world of mathematics emerges the concept of excluded values, intriguing boundaries that shape the realm of functions. These elusive values, much like forbidden doors, prevent certain operations from being performed, leaving mathematicians with clues to unlock their significance.

Causes of Excluded Values: A Journey into Mathematical Mysteries

Lurking in the shadows of mathematical operations, excluded values arise from treacherous terrains. Division by zero stands as a formidable sentinel, guarding against the undefined abyss. When a function’s denominator dares to vanish, the path forward becomes treacherous, leading to immeasurable chaos.

Venturing further, the enigmatic world of negative numbers unveils another realm of excluded values. Square roots of negative numbers dwell in this enigmatic realm, whispering tales of mathematical impossibility. Their elusive nature challenges our intuition, forcing us to delve deeper into the fabric of mathematical understanding.

Related Concepts: Asymptotes and Holes, the Guardians of Function Graphs

Excluded values have a profound impact on the way function graphs behave. Asymptotes, like towering peaks, emerge from the excluded values, marking the limits beyond which the graph cannot traverse. Holes, on the other hand, resemble tiny gaps in the graph, hinting at points where the function does not exist due to excluded values.

Unraveling the Mystery of Excluded Values: A Step-by-Step Guide

Demystifying excluded values requires a systematic approach. Begin by examining the function, seeking out expressions that involve division by zero or square roots of negative numbers. Next, set these expressions equal to zero and solve for potential excluded values. Finally, check the domain of the function, ensuring that the excluded values lie outside its permissible range.

An Illustrative Example: Shedding Light on Excluded Values

Consider the function:

f(x) = (x - 2) / (x + 1)

Step 1: Examination

Division by zero occurs when x = -1.

Step 2: Setting Expression Equal to Zero

(x – 2) / (x + 1) = 0

x – 2 = 0
x = 2
Excluded value: x = 2

Step 3: Domain Check

The domain of the function is all real numbers except x = -1. Since x = 2 is not in the domain, it is an excluded value.

Asymptotes: Discuss how excluded values can create vertical asymptotes

Excluded Values: Navigating Mathematical Exceptions

Like curious explorers venturing into uncharted territories, mathematicians encounter mathematical functions that behave in puzzling ways. Along this journey, they stumble upon excluded values, enigmatic boundaries that challenge our intuition and invite us to delve deeper into the world of mathematics.

Unveiling the Enigma of Excluded Values

In the realm of mathematics, excluded values are numbers that render a function undefined or meaningless. They arise due to mathematical operations that, when applied to certain inputs, produce nonsensical results. For instance, dividing any number by zero is like asking, “How many times can nothing go into something?” Similarly, taking the square root of a negative number leads to a mathematical paradox, as there is no real number that, when multiplied by itself, yields a negative result.

Causes of Mathematical Mischief

Excluded values originate from various mathematical operations, including:

• Division by zero: When the denominator of a fraction becomes zero, we encounter division by zero, a forbidden operation in mathematics.
• Square roots of negative numbers: Negative numbers cannot have real square roots. This is because the square of any real number is always positive.
• Logarithms of negative numbers: Logarithms are only defined for positive numbers. When the argument of a logarithm is negative, we encounter an excluded value.

Related Concepts: Asymptotes and Holes

Excluded values can have profound implications on the behavior of functions. Consider the function (f(x) = \frac{1}{x-2}). At (x = 2), the function is undefined due to division by zero, making it an excluded value. This excluded value gives rise to a vertical asymptote at (x = 2), where the function approaches infinity as (x) approaches 2 from either side.

In other scenarios, excluded values can cause holes in function graphs. For instance, the function (f(x) = \sqrt{x-1}) is not defined for (x < 1). This excluded value creates a hole in the function graph at (x = 1), where the function is undefined.

Uncovering Excluded Values: A Step-by-Step Guide

Finding excluded values requires a methodical approach. Follow these steps:

1. Examine the function: Identify any expressions involving division by zero or square roots of negative numbers.
2. Set expressions equal to zero: Solve these expressions to find potential excluded values.
3. Check the domain: Excluded values must fall outside the domain of the function.

Example: Unveiling the Excluded Value of f(x) = \frac{x+1}{x-3}

Let’s put our detective skills to the test with the function (f(x) = \frac{x+1}{x-3}).

• Step 1: The expression involving division by zero is ((x-3)). Setting it equal to zero, we get (x = 3).
• Step 2: The excluded value is (x = 3).
• Step 3: The domain of the function is all real numbers except for (x = 3).

Therefore, the excluded value of the function (f(x) = \frac{x+1}{x-3}) is (x = 3).

Understanding Excluded Values: Bringing Clarity to Mathematical Landscapes

In the realm of mathematics, excluded values play a pivotal role in defining the boundaries of functions. These are values that cannot be plugged into a function, making them non-permissible within its domain. Excluded values stem from operations that are undefined or produce complex results.

Causes of Excluded Values:

Common culprits behind excluded values include:

• Division by Zero: Attempts to divide by zero lead to infinite outcomes, rendering the function undefined.
• Square Roots of Negative Numbers: Square roots of negative numbers result in imaginary numbers. As most functions operate in the domain of real numbers, these values are excluded.
• Logarithms of Negative Numbers: Logarithms of negative numbers yield negative exponents, which are not permissible in the real number system.

Related Concepts:

Excluded values have a profound impact on function graphs. They create:

• Asymptotes: When a function approaches an excluded value as an input, its output becomes infinite, forming a vertical asymptote on the graph.
• Holes: Certain excluded values can create small gaps or discontinuities in function graphs, known as holes. These occur when the function is undefined at a specific point but defined in the surrounding domain.

Finding Excluded Values:

To identify excluded values:

1. Examine the Function: Inspect the function for expressions involving division by zero or square roots of negative numbers.
2. Set Expressions Equal to Zero: Equate these expressions to zero and solve for potential excluded values.
3. Check the Domain: Exclude values outside the function’s domain, ensuring that the remaining values satisfy the function’s requirements.

Example:

Consider the function:

f(x) = (x + 2) / (x - 1)

1. Examine the Function: The function has a term “x – 1” in the denominator, raising concerns about potential division by zero.
2. Set Expression Equal to Zero: Setting “x – 1” to zero yields x = 1, indicating a potential excluded value.
3. Check the Domain: There are no domain restrictions for this function, so x = 1 remains the only excluded value.

Excluded values are essential constructs in mathematics, defining the boundaries of functions and influencing their graphical representations. Understanding these concepts empowers us to navigate mathematical landscapes with greater precision and clarity.

Excluded Values: Uncovering the Mathematical Secrets

In the realm of mathematics, we encounter a fascinating concept known as excluded values. These are values that lie outside the permissible range of a function, creating boundaries and shaping the behavior of the graph.

Step 1: Examine the Function

To unveil the secrets of excluded values, we begin by scrutinizing the function. We seek out expressions that could potentially cause trouble. Two culprits to watch for are:

• Division by Zero: When a function involves dividing by a variable or expression that can equal zero, it raises a red flag. Zero, as we all know, is a forbidden denominator that leads to mathematical chaos.

• Square Roots of Negative Numbers: We venture into uncharted territory when we encounter expressions involving the square roots of negative numbers. Such quantities are unknown within the realm of real numbers, calling for a different approach.

Identifying Expressions That Might Lead to Excluded Values

By meticulously examining the function, we can pinpoint expressions that harbor potential excluded values. They may appear as:

• “`
  f(x) = 1/(x – 2)

Division by zero occurs when ```x - 2 = 0```, resulting in ```x = 2```. Voilà, we've found a potential excluded value!

- ```
f(x) = √(x - 3)

The square root of a negative number arises when x - 3 < 0, translating to x < 3. So, values less than 3 might be excluded.

By uncovering these potential excluded values, we lay the foundation for a deeper understanding of the function’s behavior and the secrets it holds.

Excluded Values: Unlocking Mathematical Boundaries

In the tapestry of mathematics, excluded values represent boundaries where certain operations cannot tread. These values, like forbidden fruits, hold the power to disrupt the flow of calculations, creating ripples that extend far beyond their initial impact.

The Genesis of Excluded Values: Where Mathematics Draws the Line

Excluded values arise when mathematical expressions encounter insurmountable obstacles. The most common culprits are:

• Division by zero: The magical number zero, when placed in the denominator, renders division impossible, leaving us in a state of mathematical limbo.

• Square roots of negative numbers: The realm of imaginary numbers, where the square of -1 equals -1, remains elusive to our standard operations.

• Logarithms of negative numbers: Logarithms, the inverses of exponents, only make sense for positive numbers, leaving their negative counterparts in the mathematical wilderness.

Asymptotes and Holes: The Scars of Excluded Values

Excluded values have a profound impact on the graphs of functions. They can create vertical asymptotes, where the graph approaches infinity but never quite reaches it, like a tantalizing mirage. They can also lead to holes, where the graph has a sudden break, like a missing piece in a puzzle.

Detecting Excluded Values: The Mathematical Detective’s Guide

To uncover the hidden presence of excluded values, we employ a three-step strategy:

1. Examine the Function: Our investigation begins with a careful perusal of the function’s algebraic expression. We seek out any expressions involving division by zero or square roots of negative numbers.

2. Set Expressions Equal to Zero: A crucial step involves setting these expressions equal to zero and solving for potential excluded values. These values represent the points where the function encounters its mathematical roadblocks.

3. Check the Domain: Finally, we don’t forget to consult the domain of the function. This defines the range of values for which the function is valid, and any values outside this domain are automatically excluded.

A Real-World Example: Unveiling the Secrets of an Expression

Let’s put our detective skills to the test. Consider the function:

f(x) = 1 / (x - 2) + sqrt(x + 3)

Step 1: Examining the Function

Our keen eyes spot two potential troublemakers: division by zero in the first term and a square root of a negative number in the second term.

Step 2: Setting Expressions Equal to Zero

• For the first term, we set 1 / (x – 2) = 0. Solving for x, we find x = 2.

• For the second term, we set sqrt(x + 3) = 0. However, we encounter a snag: the square root of a negative number cannot exist in our number system. This means there are no excluded values due to the square root.

Step 3: Checking the Domain

Since the function involves division, its domain is all real numbers except x = 2.

Therefore, the only excluded value for this function is x = 2.

Excluded Values: The Mathematical Enigma

In the realm of mathematics, there exist certain values that are deemed forbidden, akin to outcasts in a harmonious society. These values, known as excluded values, are those that cause expressions or functions to become undefined or produce nonsensical results. Understanding excluded values is crucial for navigating the mathematical landscape with confidence.

Unraveling the Origins of Excluded Values

Excluded values arise from mathematical operations that encounter hurdles, like division by zero or attempting to find the square root of a negative number. These situations create scenarios where the rules of mathematics break down and functions go haywire. Let’s explore some common causes:

• Division by Zero: When a number is divided by zero, the result becomes infinite or indeterminate, a mathematical no-go zone.

• Square Roots of Negative Numbers: The square root of a negative number leads us into the realm of imaginary numbers, which are beyond the scope of everyday mathematics.

• Logarithms of Negative Numbers: The logarithm of a negative number is undefined because the logarithmic function is only defined for positive numbers.

Asymptotes and Holes: Visualizing Excluded Values

Excluded values can manifest themselves in function graphs in fascinating ways:

• Asymptotes: Excluded values can create vertical asymptotes in the graph of a function. These asymptotes mark boundaries where the function approaches infinity without ever touching it.

• Holes: In some cases, excluded values can result in holes in the graph. These holes represent points where the function is undefined and cannot be evaluated.

Embarking on the Quest for Excluded Values

Finding excluded values is a process that requires a keen eye for detail:

1. Examine the Function: Scrutinize the mathematical expression or function to identify any instances of division by zero or square roots of negative numbers.

2. Set Expressions Equal to Zero: If division by zero or square roots of negative numbers are present, set those expressions equal to zero and solve for the potential excluded values.

3. Check the Domain: Recall the domain of the function, which specifies the set of valid input values. Exclude any values that lie outside the domain.

A Practical Example: Navigating Excluded Values

Let’s consider the function f(x) = (x - 2) / (x^2 - 4).

• Step 1: Examining the Function: Inspecting the function, we notice that division by zero could occur if x^2 - 4 equals zero.

• Step 2: Setting Expressions Equal to Zero: Solving x^2 - 4 = 0 yields x = ±2.

• Step 3: Checking the Domain: The domain of the function is all real numbers except for x = ±2. Thus, our excluded values are -2 and 2.

By understanding the concept of excluded values, we equip ourselves with the tools to navigate the treacherous waters of mathematics. These outcasts may seem like obstacles, but with the right approach, they become stepping stones to mathematical enlightenment.

Understanding Excluded Values in Mathematics

What are Excluded Values?

In mathematics, excluded values refer to specific input values that make mathematical expressions undefined. These values are excluded from the domain of the function, as they lead to division by zero or other invalid operations.

Common examples of excluded values include:

• Division by Zero: Any number divided by zero is undefined, resulting in the excluded value of zero.
• Square Roots of Negative Numbers: The square root of a negative number is not a real number, so these values are excluded.
• Logarithms of Negative Numbers: It’s impossible to take the logarithm of a negative number, leading to the exclusion of negative values.

Causes of Excluded Values

Excluded values arise due to certain mathematical operations that produce undefined results. These include:

• Division by Zero
• Square Roots of Negative Numbers
• Logarithms of Negative Numbers

Related Concepts: Asymptotes and Holes

Excluded values can have a significant impact on the graph of a function. They can create:

• Vertical Asymptotes: When a function approaches infinity as it gets closer to an excluded value, it creates a vertical asymptote.
• Holes: If a function has a removable discontinuity at an excluded value, it results in a hole in the graph.

Finding Excluded Values

Finding excluded values involves a three-step process:

1. Examine the Function: Look for expressions involving division by zero or square roots of negative numbers.
2. Set Expressions Equal to Zero: Solve the expressions to find potential excluded values.
3. Check the Domain: Exclude any values outside the domain of the function.

Example: Finding Excluded Values

Consider the function: f(x) = 1/(x-2)

Step 1: Examine the Function

The expression involves division by zero, so we set:

x-2 = 0

Step 2: Set Expressions Equal to Zero

Solving for x, we get:

x = 2

Step 3: Check the Domain

The function’s domain excludes x=2, as it produces an excluded value of zero.

Therefore, the excluded value for this function is x=2.

Unveiling the Enigma of Excluded Values in Mathematics

Excluded values are mathematical values that are forbidden from entering specific expressions or functions, often leading to undefined results. Understanding these enigmatic values is crucial for comprehending the behavior of functions and avoiding mathematical pitfalls.

What are Excluded Values?

In mathematics, excluded values are those that make an expression or function undefined. They can arise due to various reasons, such as:

• Division by zero: When the denominator of a fraction becomes zero, the result is undefined.
• Square roots of negative numbers: Attempting to take the square root of a negative number yields an imaginary result, which is not part of the real number system.
• Logarithms of negative numbers: Logarithms are defined only for positive numbers, and negative values result in undefined expressions.

Causes of Excluded Values

The most common causes of excluded values are:

Division by Zero

When dividing one number by another, the denominator cannot be zero. If it is, the result is undefined because dividing by zero means attempting to split a quantity into an infinite number of equal parts.

Square Roots of Negative Numbers

The square root function is defined only for non-negative numbers. Extracting the square root of a negative number would require the existence of a number that, when multiplied by itself, yields a negative result, which is impossible within the realm of real numbers.

Logarithms of Negative Numbers

Logarithms are used to find the exponent to which a base number must be raised to produce a given result. However, the base number must be positive, and negative values lead to undefined expressions.

Related Concepts

Excluded values can impact functions in several ways:

Asymptotes

Excluded values can create vertical asymptotes in function graphs. These are vertical lines where the function approaches infinity or negative infinity, indicating that the value at the asymptote is undefined.

Holes

Alternatively, excluded values can also create holes in function graphs. These are points where the function is undefined, but the graph remains continuous on either side of the hole.

Finding Excluded Values

To find excluded values, follow these steps:

1. Examine the Function: Identify expressions involving division by zero, square roots of negative numbers, or logarithms of negative numbers.
2. Set Expressions Equal to Zero: Solve for the values that make these expressions zero.
3. Check the Domain: Ensure that the excluded values lie outside the domain of the function, which is the set of all valid input values.

Example

Consider the function f(x) = (x-2)/(x^2 - 3x + 2).

Step 1: Examine the Function

• Division by zero: x^2 - 3x + 2 cannot equal zero. Solve: x^2 - 3x + 2 = 0.
  • Solutions: x = 1, 2
• Square roots of negative numbers: None

Step 2: Set Expressions Equal to Zero

• Set x^2 - 3x + 2 to zero: x = 1, 2

Step 3: Check the Domain

The domain is all real numbers except x = 1, 2.

Therefore, the excluded values are **x = 1, 2**.