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Find the inverse of the function: h(x) = ² + 2

Find the inverse of the function: h(x) = ² + 2

Algebra and Trigonometry (6th Edition)
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: Find the Inverse of a Function** Given the function: \[ h(x) = \frac{3}{x} + 2 \] **Select the correct inverse function from the options below:** **A.** \[ \frac{2}{x-2} \] **B.** \[ \frac{3}{x+2} \] **C.** \[ \frac{2}{x+3} \] **D.** \[ \frac{3}{x-2} \] **E.** \[ \frac{2}{x-3} \] To determine the inverse of a function, we follow these steps: 1. Replace \( h(x) \) with \( y \). 2. Swap \( x \) and \( y \). 3. Solve for \( y \) to find the inverse function \( h^{-1}(x) \). For the given function \( h(x) = \frac{3}{x} + 2 \): 1. Start with \( y = \frac{3}{x} + 2 \). 2. Swap \( x \) and \( y \) to get \( x = \frac{3}{y} + 2 \). 3. Solve for \( y \): \[ x - 2 = \frac{3}{y} \] \[ y = \frac{3}{x - 2} \] Thus, the inverse function \( h^{-1}(x) \) is \( \frac{3}{x - 2} \). The correct answer is **D: \(\frac{3}{x-2}\)**.
Transcribed Image Text:**Problem: Find the Inverse of a Function** Given the function: \[ h(x) = \frac{3}{x} + 2 \] **Select the correct inverse function from the options below:** **A.** \[ \frac{2}{x-2} \] **B.** \[ \frac{3}{x+2} \] **C.** \[ \frac{2}{x+3} \] **D.** \[ \frac{3}{x-2} \] **E.** \[ \frac{2}{x-3} \] To determine the inverse of a function, we follow these steps: 1. Replace \( h(x) \) with \( y \). 2. Swap \( x \) and \( y \). 3. Solve for \( y \) to find the inverse function \( h^{-1}(x) \). For the given function \( h(x) = \frac{3}{x} + 2 \): 1. Start with \( y = \frac{3}{x} + 2 \). 2. Swap \( x \) and \( y \) to get \( x = \frac{3}{y} + 2 \). 3. Solve for \( y \): \[ x - 2 = \frac{3}{y} \] \[ y = \frac{3}{x - 2} \] Thus, the inverse function \( h^{-1}(x) \) is \( \frac{3}{x - 2} \). The correct answer is **D: \(\frac{3}{x-2}\)**.
**Graph Analysis: Function Verification** Text: The following graph is the graph of a function. [Graph Description]: - The graph displays a curve on a Cartesian plane. - The curve passes through the origin and splits into two branches: - One branch is in the first quadrant, curving upwards to the right. - The other branch is in the third quadrant, curving downwards to the left. Select one: - True - False
Transcribed Image Text:**Graph Analysis: Function Verification** Text: The following graph is the graph of a function. [Graph Description]: - The graph displays a curve on a Cartesian plane. - The curve passes through the origin and splits into two branches: - One branch is in the first quadrant, curving upwards to the right. - The other branch is in the third quadrant, curving downwards to the left. Select one: - True - False
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