Composition of Functions with Given Functions

Oct 13, 2025 by windailys.com 112 views

Introduction

In mathematics, a composition of functions is a way of combining two or more functions to create a new function. This is a fundamental concept in algebra and calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of composition of functions, and we will use the given functions $s(x) = 2 - x^2$ and $t(x) = 3x$ to find the value of $(s \circ t)(-7)$ .

What is a Composition of Functions?

A composition of functions is a way of combining two or more functions to create a new function. Given two functions $f(x)$ and $g(x)$ , the composition of $f$ and $g$ is denoted by $f \circ g$ and is defined as:

(f \circ g)(x) = f(g(x))

In other words, we first apply the function $g$ to the input $x$ , and then we apply the function $f$ to the result.

Example: Composition of Two Functions

Let's consider two functions $f(x) = 2x$ and $g(x) = x^2$ . We can find the composition of $f$ and $g$ as follows:

(f \circ g)(x) = f(g(x)) = f(x^2) = 2(x^2) = 2x^2

Composition of Functions with Given Functions

Now, let's consider the given functions $s(x) = 2 - x^2$ and $t(x) = 3x$ . We want to find the value of $(s \circ t)(-7)$ .

To do this, we need to apply the function $t$ to the input $-7$ , and then apply the function $s$ to the result.

Step 1: Apply the Function t to the Input -7

First, we apply the function $t$ to the input $-7$ :

t(-7) = 3(-7) = -21

Step 2: Apply the Function s to the Result

Next, we apply the function $s$ to the result:

s(-21) = 2 - (-21)^2 = 2 - 441 = -439

Conclusion

Therefore, the value of $(s \circ t)(-7)$ is equivalent to $-439$ .

Answer

The correct answer is:

A. $-439$

Discussion

The composition of functions is a powerful tool in mathematics, and it has numerous applications in various fields. In this article, we used the given functions $s(x) = 2 - x^2$ and $t(x) = 3x$ to find the value of $(s \circ t)(-7)$ . We applied the function $t$ to the input $-7$ , and then applied the function $s$ to the result. The final answer is $-439$ .

Tips and Tricks

When working with composition of functions, make sure to apply the functions in the correct order.
Use the definition of composition of functions to find the value of $(f \circ g)(x)$ .
Practice, practice, practice! The more you practice, the more comfortable you will become with composition of functions.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill

Glossary

Composition of functions: A way of combining two or more functions to create a new function.
Function: A relation between a set of inputs and a set of possible outputs.
Input: The value that is fed into a function.
Output: The value that is produced by a function.
Composition of Functions: Q&A =============================

Introduction

In our previous article, we explored the concept of composition of functions and used the given functions $s(x) = 2 - x^2$ and $t(x) = 3x$ to find the value of $(s \circ t)(-7)$ . In this article, we will answer some frequently asked questions about composition of functions.

Q&A

Q: What is the difference between a function and a composition of functions?

A: A function is a relation between a set of inputs and a set of possible outputs. A composition of functions is a way of combining two or more functions to create a new function.

Q: How do I find the value of $(f \circ g)(x)$ ?

A: To find the value of $(f \circ g)(x)$ , you need to apply the function $g$ to the input $x$ , and then apply the function $f$ to the result.

Q: What is the order of operations when working with composition of functions?

A: When working with composition of functions, you need to apply the functions in the correct order. This means that you need to apply the function on the left first, and then apply the function on the right.

Q: Can I use composition of functions to solve equations?

A: Yes, you can use composition of functions to solve equations. For example, if you have an equation like $f(g(x)) = k$ , you can use composition of functions to find the value of $x$ .

Q: How do I know if a function is a composition of functions?

A: To determine if a function is a composition of functions, you need to check if it can be written as a combination of two or more functions. If it can be written in this way, then it is a composition of functions.

Q: Can I use composition of functions to find the inverse of a function?

A: Yes, you can use composition of functions to find the inverse of a function. For example, if you have a function $f(x)$ and you want to find its inverse, you can use composition of functions to find the inverse.

Q: What are some common mistakes to avoid when working with composition of functions?

A: Some common mistakes to avoid when working with composition of functions include:

Not applying the functions in the correct order
Not using the correct notation for composition of functions
Not checking if a function is a composition of functions before using it

Tips and Tricks

When working with composition of functions, make sure to apply the functions in the correct order.
Use the definition of composition of functions to find the value of $(f \circ g)(x)$ .
Practice, practice, practice! The more you practice, the more comfortable you will become with composition of functions.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill

Glossary

Composition of functions: A way of combining two or more functions to create a new function.
Function: A relation between a set of inputs and a set of possible outputs.
Input: The value that is fed into a function.
Output: The value that is produced by a function.
Inverse function: A function that undoes the action of another function.

Practice Problems

Find the value of $(f \circ g)(x)$ if $f(x) = 2x$ and $g(x) = x^2$ .
Find the value of $(f \circ g)(x)$ if $f(x) = x^2$ and $g(x) = 2x$ .
Find the inverse of the function $f(x) = 2x$ .
Find the inverse of the function $f(x) = x^2$ .

Answer Key

$(f \circ g)(x) = 4x^2$
$(f \circ g)(x) = 4x^4$
The inverse of the function $f(x) = 2x$ is $f^{-1}(x) = \frac{x}{2}$ .
The inverse of the function $f(x) = x^2$ is $f^{-1}(x) = \sqrt{x}$ .