**Problem Statement:**If \( g(x) = 3x - 5 \) and \( h(x) = (f \circ g)(x) = 9x^2 - 30x + 25 \), then \( f(x) = \boxed{3x - 5} \ \textcolor{red}{\text{✘}} \).**Objective:**Using \( g(x) \) and \( f(x) \), determine \( (g \circ f)(x) \).\( (g \circ f)(x) = \boxed{9x - 20} \ \textcolor{red}{\text{✘}} \).**Explanation:**1. **Function Definitions:** - \( g(x) \) is defined as \( 3x - 5 \). - The composition of functions \( h(x) = (f \circ g)(x) \) is given as a quadratic expression \( 9x^2 - 30x + 25 \).2. **Incorrect Solutions Identified:** - The proposed solution for \( f(x) \) was incorrectly stated as \( 3x - 5 \). - The solution for \( (g \circ f)(x) \) was incorrectly determined to be \( 9x - 20 \). This exercise explores the errors made in finding the correct composite functions. Understanding how to correctly compute compositions is crucial for various applications in mathematics. **How to Transform the Graph of \( g(x) = \frac{13}{14} |x - 6| + 3 \) from \( f(x) = |x| \):**1. **Horizontal Shift:** - The graph of \( g(x) \) is obtained by shifting the graph of \( f(x) \) 6 units to the **right**.2. **Reflection:** - The graph of \( g(x) \) **is not** a reflection of the graph of \( f(x) \) about the x-axis.3. **Vertical Shrink/Stretch:** - The graph of \( g(x) \) is the graph of \( f(x) \) **shrunk** by a factor of \( \frac{13}{14} \).4. **Vertical Shift:** - The graph of \( g(x) \) is shifted **up** by 3 units.

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Answered: If g(x) = 3x - 5 and h(x) = (fog)(x) =…

If g(x) = 3x - 5 and h(x) = (fog)(x) = 9x² 30x + 25, then f(x) = 3x - 5 Using g(x) and f(x), determine (g o f)(x). (gof)(x) = 9x - 20 X x.

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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Question

**Problem Statement:**

If \( g(x) = 3x - 5 \) and \( h(x) = (f \circ g)(x) = 9x^2 - 30x + 25 \), then \( f(x) = \boxed{3x - 5} \ \textcolor{red}{\text{✘}} \).

**Objective:**

Using \( g(x) \) and \( f(x) \), determine \( (g \circ f)(x) \).

\( (g \circ f)(x) = \boxed{9x - 20} \ \textcolor{red}{\text{✘}} \).

**Explanation:**

1. **Function Definitions:**

- \( g(x) \) is defined as \( 3x - 5 \).
- The composition of functions \( h(x) = (f \circ g)(x) \) is given as a quadratic expression \( 9x^2 - 30x + 25 \).

2. **Incorrect Solutions Identified:**

- The proposed solution for \( f(x) \) was incorrectly stated as \( 3x - 5 \).
- The solution for \( (g \circ f)(x) \) was incorrectly determined to be \( 9x - 20 \).

This exercise explores the errors made in finding the correct composite functions. Understanding how to correctly compute compositions is crucial for various applications in mathematics.

**How to Transform the Graph of \( g(x) = \frac{13}{14} |x - 6| + 3 \) from \( f(x) = |x| \):**

1. **Horizontal Shift:**
- The graph of \( g(x) \) is obtained by shifting the graph of \( f(x) \) 6 units to the **right**.

2. **Reflection:**
- The graph of \( g(x) \) **is not** a reflection of the graph of \( f(x) \) about the x-axis.

3. **Vertical Shrink/Stretch:**
- The graph of \( g(x) \) is the graph of \( f(x) \) **shrunk** by a factor of \( \frac{13}{14} \).

4. **Vertical Shift:**
- The graph of \( g(x) \) is shifted **up** by 3 units.

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