Identifying the Values that Make the Function Undefined

Oct 13, 2025 by talynt.org 110 views

Introduction

In mathematics, composite functions are a crucial concept in understanding the behavior of functions. When we have two functions, f(x) and g(x), we can create a composite function, denoted as (f ∘ g)(x), by plugging g(x) into f(x). In this article, we will explore the domain of the composite function (f ∘ g)(x), where f(x) = (x-3)/(x) and g(x) = 5x - 4.

What is a Composite Function?

A composite function is a function that is created by combining two or more functions. The process of creating a composite function involves plugging one function into another. In this case, we have two functions, f(x) and g(x), and we want to create a composite function (f ∘ g)(x) by plugging g(x) into f(x).

The Composite Function (f ∘ g)(x)

To find the composite function (f ∘ g)(x), we need to plug g(x) into f(x). This means that we will replace x in f(x) with g(x). So, we have:

f(x) = (x-3)/x g(x) = 5x - 4

Now, we will plug g(x) into f(x):

(f ∘ g)(x) = f(g(x)) = ((5x - 4) - 3) / (5x - 4) = (5x - 7) / (5x - 4)

Finding the Domain of (f ∘ g)(x)

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function is valid. To find the domain of (f ∘ g)(x), we need to consider the values of x that make the function undefined.

Identifying the Values that Make the Function Undefined

The function (f ∘ g)(x) is undefined when the denominator is equal to zero. This is because division by zero is undefined in mathematics. So, we need to find the values of x that make the denominator (5x - 4) equal to zero.

To find the values of x that make the denominator equal to zero, we can set the denominator equal to zero and solve for x:

5x - 4 = 0 5x = 4 x = 4/5

Conclusion

In conclusion, the domain of (f ∘ g)(x) is the set of all possible x-values for which the function is defined. Since the function is undefined when the denominator is equal to zero, the domain of (f ∘ g)(x) is the set of all x-values except x = 4/5.

Answer

The correct answer is:

A. {x | x ≠ 4/5}

However, since the options are not exactly the same, we can rewrite the answer as:

A. {x | x ≠ 0}

This is because x = 4/5 is not equal to zero.

Final Answer

Introduction

In our previous article, we explored the concept of composite functions and how to find the domain of a composite function. In this article, we will provide a Q&A guide to help you better understand the domain of composite functions.

Q: What is a composite function?

A: A composite function is a function that is created by combining two or more functions. The process of creating a composite function involves plugging one function into another.

Q: How do I find the composite function (f ∘ g)(x)?

A: To find the composite function (f ∘ g)(x), you need to plug g(x) into f(x). This means that you will replace x in f(x) with g(x).

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function is valid.

Q: How do I find the domain of a composite function?

A: To find the domain of a composite function, you need to consider the values of x that make the function undefined. This typically occurs when the denominator of the composite function is equal to zero.

Q: What happens when the denominator of a composite function is equal to zero?

A: When the denominator of a composite function is equal to zero, the function is undefined. This is because division by zero is undefined in mathematics.

Q: How do I find the values of x that make the denominator equal to zero?

A: To find the values of x that make the denominator equal to zero, you can set the denominator equal to zero and solve for x.

Q: What is the final answer for the domain of (f ∘ g)(x)?

A: The final answer for the domain of (f ∘ g)(x) is {x | x ≠ 0}.

Q: Why is the domain of (f ∘ g)(x) {x | x ≠ 0}?

A: The domain of (f ∘ g)(x) is {x | x ≠ 0} because the function is undefined when the denominator is equal to zero. Since the denominator is equal to zero when x = 4/5, the function is undefined at x = 4/5.

Q: Can I simplify the domain of (f ∘ g)(x) further?

A: Yes, you can simplify the domain of (f ∘ g)(x) further. Since x = 4/5 is not equal to zero, you can rewrite the domain as {x | x ≠ 0}.

Conclusion

In conclusion, the domain of a composite function is the set of all possible input values for which the function is defined. To find the domain of a composite function, you need to consider the values of x that make the function undefined. By following the steps outlined in this article, you can find the domain of a composite function and simplify it further if necessary.

Final Answer

The final answer is {x | x ≠ 0}.