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1 Find the inverse of each of the following functions. f: x 6x + 5, x ER x +4 5 a b f:x→ cf: x4-2x, x = R 2x + 7 d f:x→ 3 e f i , ER f: x2x³ +5, x € R 1 f: xH +4, ER and x 0. x f: x→ 5 -, ER g f:x→ x-1 h f: x (x + 2)² + 7, x = Rand x > -2 3 (2x 3)² -5, x R and x > 2 *, €Randr #1 j f: x x² - 6x, x = R and x > 3 2 A function is called self-inverse if f(x) = f(x) for all x in the domain. Show that the following functions are self-inverse. a f: x5-x, x ER b f:xx, x ER c f:x+ 4 -, ER anda #0 x

1 Find the inverse of each of the following functions. f: x 6x + 5, x ER x +4 5 a b f:x→ cf: x4-2x, x = R 2x + 7 d f:x→ 3 e f i , ER f: x2x³ +5, x € R 1 f: xH +4, ER and x 0. x f: x→ 5 -, ER g f:x→ x-1 h f: x (x + 2)² + 7, x = Rand x > -2 3 (2x 3)² -5, x R and x > 2 *, €Randr #1 j f: x x² - 6x, x = R and x > 3 2 A function is called self-inverse if f(x) = f(x) for all x in the domain. Show that the following functions are self-inverse. a f: x5-x, x ER b f:xx, x ER c f:x+ 4 -, ER anda #0 x

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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# Inverse Functions ## 1. Determine the Inverse Find the inverse of each of the following functions. - **a**: \( f : x \mapsto 2x - 3, x \in \mathbb{R} \) - **b**: \( f : x \mapsto 4 - x^2, x \in \mathbb{R} \) - **c**: \( f : x \mapsto \dfrac{1}{2x}, x \in \mathbb{R} \) - **d**: \( f : x \mapsto 3, x \in \mathbb{R} \) - **e**: \( f : x \mapsto 2x^3 + 5, x \in \mathbb{R} \) - **f**: \( f : x \mapsto \dfrac{1}{x} + 4, x \in \mathbb{R} \) and \( x \neq 0 \) - **g**: \( f : x \mapsto \dfrac{5}{x - 1}, x \in \mathbb{R} \) and \( x \neq 1 \) - **h**: \( f : x \mapsto (x + 2)^2 + 7, x \in \mathbb{R} \) and \( x \geq -2 \) - **i**: \( f : x \mapsto (2x - 3)^2 - 5, x \in \mathbb{R} \) and \( x \geq \dfrac{3}{2} \) - **j**: \( f : x \mapsto x^2 - 6x, x \in \mathbb{R} \) and \( x \geq 3 \) ## 2. Self-Inverse Functions A function is called **self-inverse** if \( f(x) = f^{-1}(x) \) for all \( x \) in the domain. Show that the following functions are self-inverse. - **a**: \( f : x \mapsto 5 - x, x \in \mathbb{R} \) - **b**: \( f : x \mapsto -x, x \in \mathbb{R
Transcribed Image Text:# Inverse Functions ## 1. Determine the Inverse Find the inverse of each of the following functions. - **a**: \( f : x \mapsto 2x - 3, x \in \mathbb{R} \) - **b**: \( f : x \mapsto 4 - x^2, x \in \mathbb{R} \) - **c**: \( f : x \mapsto \dfrac{1}{2x}, x \in \mathbb{R} \) - **d**: \( f : x \mapsto 3, x \in \mathbb{R} \) - **e**: \( f : x \mapsto 2x^3 + 5, x \in \mathbb{R} \) - **f**: \( f : x \mapsto \dfrac{1}{x} + 4, x \in \mathbb{R} \) and \( x \neq 0 \) - **g**: \( f : x \mapsto \dfrac{5}{x - 1}, x \in \mathbb{R} \) and \( x \neq 1 \) - **h**: \( f : x \mapsto (x + 2)^2 + 7, x \in \mathbb{R} \) and \( x \geq -2 \) - **i**: \( f : x \mapsto (2x - 3)^2 - 5, x \in \mathbb{R} \) and \( x \geq \dfrac{3}{2} \) - **j**: \( f : x \mapsto x^2 - 6x, x \in \mathbb{R} \) and \( x \geq 3 \) ## 2. Self-Inverse Functions A function is called **self-inverse** if \( f(x) = f^{-1}(x) \) for all \( x \) in the domain. Show that the following functions are self-inverse. - **a**: \( f : x \mapsto 5 - x, x \in \mathbb{R} \) - **b**: \( f : x \mapsto -x, x \in \mathbb{R
### Inverse Functions 1. **Find the inverse of each of the following functions:** a. \( f : x \mapsto 6x + 5, \, x \in \mathbb{R} \) b. \( f : x \mapsto \frac{x + 4}{5}, \, x \in \mathbb{R} \) c. \( f : x \mapsto 4 - 2x, \, x \in \mathbb{R} \) d. \( f : x \mapsto \frac{2x + 7}{3}, \, x \in \mathbb{R} \) e. \( f : x \mapsto 2x^3 + 5, \, x \in \mathbb{R} \) f. \( f : x \mapsto \frac{1}{x} + 4, \, x \in \mathbb{R} \text{ and } x \neq 0 \) g. \( f : x \mapsto \frac{5}{x - 1}, \, x \in \mathbb{R} \text{ and } x \neq 1 \) h. \( f : x \mapsto (x + 2)^2 + 7, \, x \in \mathbb{R} \text{ and } x \geq -2 \) i. \( f : x \mapsto (2x - 3)^2 - 5, \, x \in \mathbb{R} \text{ and } x \geq \frac{3}{2} \) j. \( f : x \mapsto x^2 - 6x, \, x \in \mathbb{R} \text{ and } x \geq 3 \) 2. **Self-Inverse Functions** A function is called self-inverse if \( f(x) = f^{-1}(x) \) for all \( x \) in the domain. Show that the following functions are self-inverse. a. \( f : x \mapsto 5 - x, \, x \in \mathbb{R} \) b. \( f : x \mapsto -x, \, x \in \math
Transcribed Image Text:### Inverse Functions 1. **Find the inverse of each of the following functions:** a. \( f : x \mapsto 6x + 5, \, x \in \mathbb{R} \) b. \( f : x \mapsto \frac{x + 4}{5}, \, x \in \mathbb{R} \) c. \( f : x \mapsto 4 - 2x, \, x \in \mathbb{R} \) d. \( f : x \mapsto \frac{2x + 7}{3}, \, x \in \mathbb{R} \) e. \( f : x \mapsto 2x^3 + 5, \, x \in \mathbb{R} \) f. \( f : x \mapsto \frac{1}{x} + 4, \, x \in \mathbb{R} \text{ and } x \neq 0 \) g. \( f : x \mapsto \frac{5}{x - 1}, \, x \in \mathbb{R} \text{ and } x \neq 1 \) h. \( f : x \mapsto (x + 2)^2 + 7, \, x \in \mathbb{R} \text{ and } x \geq -2 \) i. \( f : x \mapsto (2x - 3)^2 - 5, \, x \in \mathbb{R} \text{ and } x \geq \frac{3}{2} \) j. \( f : x \mapsto x^2 - 6x, \, x \in \mathbb{R} \text{ and } x \geq 3 \) 2. **Self-Inverse Functions** A function is called self-inverse if \( f(x) = f^{-1}(x) \) for all \( x \) in the domain. Show that the following functions are self-inverse. a. \( f : x \mapsto 5 - x, \, x \in \mathbb{R} \) b. \( f : x \mapsto -x, \, x \in \math
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