Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Compositions of Functions
In this exercise, you are given two functions:
\[ f(x) = 3x - 2 \]
\[ g(x) = 5 - x^2 \]
You need to evaluate the following composed functions:
1. **(a) \((f \circ f)(x)\)**:
- This represents the composition of \(f\) with itself.
- To find \((f \circ f)(x)\), you substitute \(f(x)\) into \(f\).
2. **(b) \((g \circ g)(x)\)**:
- This represents the composition of \(g\) with itself.
- To find \((g \circ g)(x)\), you substitute \(g(x)\) into \(g\).
For your reference:
- An empty box is provided where you can write your answers for each composed function.
These compositions require you to substitute the output of one function into the input of the same function. The procedure for solving compositions involves the following steps:
### Solving Compositions
1. **Calculate \((f \circ f)(x)\)**:
- First, compute \(f(x)\): \( f(x) = 3x - 2 \).
- Next, substitute \(f(x)\) into itself: \( f(f(x)) = f(3x - 2) \).
- Then solve: \( f(3x - 2) = 3(3x - 2) - 2 \).
2. **Calculate \((g \circ g)(x)\)**:
- First, compute \(g(x)\): \( g(x) = 5 - x^2 \).
- Next, substitute \(g(x)\) into itself: \( g(g(x)) = g(5 - x^2) \).
- Then solve: \( g(5 - x^2) = 5 - (5 - x^2)^2 \).
Dequeen these steps to find the specific values for the composed functions. Write your final answers in the provided boxes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f4719ec-89c6-49f2-a8fd-c905fe776b01%2F8a8e13e6-ee3b-442d-a7df-08627c20d593%2Fqo2ofz6_processed.png&w=3840&q=75)
Transcribed Image Text:### Compositions of Functions
In this exercise, you are given two functions:
\[ f(x) = 3x - 2 \]
\[ g(x) = 5 - x^2 \]
You need to evaluate the following composed functions:
1. **(a) \((f \circ f)(x)\)**:
- This represents the composition of \(f\) with itself.
- To find \((f \circ f)(x)\), you substitute \(f(x)\) into \(f\).
2. **(b) \((g \circ g)(x)\)**:
- This represents the composition of \(g\) with itself.
- To find \((g \circ g)(x)\), you substitute \(g(x)\) into \(g\).
For your reference:
- An empty box is provided where you can write your answers for each composed function.
These compositions require you to substitute the output of one function into the input of the same function. The procedure for solving compositions involves the following steps:
### Solving Compositions
1. **Calculate \((f \circ f)(x)\)**:
- First, compute \(f(x)\): \( f(x) = 3x - 2 \).
- Next, substitute \(f(x)\) into itself: \( f(f(x)) = f(3x - 2) \).
- Then solve: \( f(3x - 2) = 3(3x - 2) - 2 \).
2. **Calculate \((g \circ g)(x)\)**:
- First, compute \(g(x)\): \( g(x) = 5 - x^2 \).
- Next, substitute \(g(x)\) into itself: \( g(g(x)) = g(5 - x^2) \).
- Then solve: \( g(5 - x^2) = 5 - (5 - x^2)^2 \).
Dequeen these steps to find the specific values for the composed functions. Write your final answers in the provided boxes.
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