u can use any function u like and answer these questions.. show all work **Mathematical Problems and Exercises**1. **Given a Function $ f $:** - (a) Demonstrate (through appropriate computation and not by viewing the graph) that $ f $ is one-to-one and hence has an inverse function. - (b) Determine the rule that expresses $ f^{-1} $ as a function of $ x $.2. **Given a Function $ f $:** - (a) Identify where $ f $ is increasing, decreasing, concave upward, or concave downward. - (b) Compute the area of the described region. - **Note:** Review the necessary material from the previous course to answer this question effectively!3. **Differentiation Tasks:** - (a) Utilize the method of logarithmic differentiation to find $ dy/dx $. - (b) Given $ y $, calculate its derivative. *(Note: $ y $ involves exponential functions.)*4. **Differential Equation and Derivative:** - (a) Solve a first-order, separable differential equation. - (b) Given $ y $, determine its derivative. *(Note: $ y $ involves inverse trigonometric functions.)*

Answered: 1. Given a function f, (a) show (by a…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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u can use any function u like and answer these questions..

show all work

**Mathematical Problems and Exercises**

1. **Given a Function $ f $:**
- (a) Demonstrate (through appropriate computation and not by viewing the graph) that $ f $ is one-to-one and hence has an inverse function.
- (b) Determine the rule that expresses $ f^{-1} $ as a function of $ x $.

2. **Given a Function $ f $:**
- (a) Identify where $ f $ is increasing, decreasing, concave upward, or concave downward.
- (b) Compute the area of the described region.
- **Note:** Review the necessary material from the previous course to answer this question effectively!

3. **Differentiation Tasks:**
- (a) Utilize the method of logarithmic differentiation to find $ dy/dx $.
- (b) Given $ y $, calculate its derivative. *(Note: $ y $ involves exponential functions.)*

4. **Differential Equation and Derivative:**
- (a) Solve a first-order, separable differential equation.
- (b) Given $ y $, determine its derivative. *(Note: $ y $ involves inverse trigonometric functions.)*

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$One - to - One function : - A function is said to be one - to - one if it is one - one and onto . One - one function : - A function f (x) is said to one - one if f (x) = f (y) implies x = y . Onto function : - A function f (x) is said to be onto if for every element y of codomain there always exists a element x in domain such that f (x) = y . As per the Bertleby Guideline I have to solve only first question . For answer to others resubmit the question again .$