**Question 12****Topic:** Inverse Functions**Instruction:** This is a multiple-choice question. Please select all correct answers. For which of the following functions is the inverse relation a function? Note: There is no partial credit for this question.1. **Function $ k $:** - Domain: $(-\infty, -\frac{1}{\sqrt{3}}]$ - Codomain: $\mathbb{R}$ - Function Definition: $ k(x) = x^3 - x $2. **Function $ g $:** - Domain: $[0, 1]$ - Codomain: $\mathbb{R}$ - Function Definition: $ g(x) = x^3 - x $3. **Function $ f $:** - Domain: $\mathbb{R}$ - Codomain: $\mathbb{R}$ - Function Definition: $ f(x) = x^3 - x $4. **Function $ h $:** - Domain: $[1, \infty)$ - Codomain: $\mathbb{R}$ - Function Definition: $ h(x) = x^3 - x $**Note:** Understanding which intervals make the inverse a function involves analyzing where the original function $ x^3 - x $ is one-to-one.

Name: **Question 12****Topic:** Inverse Functions**Instruction:** This is a
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Description: **Question 12****Topic:** Inverse Functions**Instruction:** This is a multiple-choice question. Please select all correct answers. For which of the following functions is the inverse relation a function? Note: There is no partial credit for this que

Answered: Multiple answer- please select all…

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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$\begin{array}{l} Which of the following functions is the inverse \\ relation a function ? \\ (1) k : (- \infty, \frac{1}{\sqrt{3}}] \to ℝ, where k (x) = x^{3} - x \\ (2) g : [0, 1] \to ℝ, where g (x) = x^{3} - x \\ (3) f : ℝ \to ℝ, where f (x) = x^{3} - x \\ (4) h : [1, \infty) \to ℝ, where h (x) = x^{3} - x \end{array}$

Step 2

$\begin{array}{l} (1) \\ \underline{Explanation :} \\ Consider a given function k (x) = x^{3} - x defined on domain \\ (- \infty, \frac{1}{\sqrt{3}}] . \\ As the polynomial k is continuous and differential on domain \\ (- \infty, \frac{1}{\sqrt{3}}] . \\ k (- \infty) = - \infty and k (\frac{1}{\sqrt{3}}) = {(\frac{1}{\sqrt{3}})}^{3} - (\frac{1}{\sqrt{3}}) = - \frac{2}{3 \sqrt{3}} . \\ hence, its range = (- \infty, - \frac{2}{3 \sqrt{3}}] \neq Co - domain (ℝ) . \\ As the function k is one - one but not onto because its range \\ is not equal to its co - domain . \\ Therefore, inverse function of k does not exist . \end{array}$

Step 3

$\begin{array}{l} (2) \\ \underline{Explanation :} \\ Consider a function g : [0, 1] \to ℝ, where g (x) = x^{3} - x . \\ As a function g is a polynomial of degree three . \\ Every continuous function map compact to compact set . \\ Hence, range of g will be compact set which is not equal \\ to set of real number (ℝ) . \\ It implies that the function g is not an onto function . Hence, \\ its inverse does not exist . \end{array}$

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