CAT Inverse Functions - Practice Questions & MCQ

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Quick Facts

2 Questions around this concept.

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MEDIUM

If $f(x)=\frac{5x-3}{2x+1}$ . Find the inverse of f(x)

$\frac{x+3}{2x-5}; x\neq5/2$

$\frac{-(x+3)}{2x-5}; x\neq5/2$

$\frac{(x+3)}{2x-5}; x\neq5/2$

$\frac{(x-3)}{2x-5}; x\neq5/2$

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Concepts Covered - 1

Inverse Functions

Introduction
Definitions
Domain and Range
Examples in Management Entrances
Properties of Inverse Functions
Tips and Tricks

Introduction

Inverse functions are an important concept in mathematics, particularly in the field of algebra and calculus. In the context of management entrance exams, understanding inverse functions is essential for solving problems related to functions and their properties.

Definitions

An inverse function of a given function f(x) is a function which, when applied to the result of the original function, gives the input value back.

In mathematical notation, if f(x) is a function and its inverse is denoted by f-1(x), then the following conditions must satisfy:

f(f-1(x)) = x
f-1(f(x)) = x

Domain and Range

The domain of an inverse function is the range of the original function, and the range of an inverse function is the domain of the original function.

For example, if f(x) = 2x + 3, then the domain of f(x) is all real numbers, and the range of f(x) is also all real numbers. The inverse function, f-1(x) = (x - 3) / 2, will have a domain and range same as that of f(x).

Examples in Management Entrances

Let's take a look at a few examples from previous management entrance exams that have tested the concept of inverse functions:

Example 1: If f(x) = 3x + 2, find the inverse function f-1(x).
Example 2: If g(x) = √(x - 4), find the inverse function g-1(x).
Example 3 : If f(x) = log3(x + 2), find the inverse function f-1(x).

Solutions for these examples:

Example 1: To find the inverse function, interchange x and y in f(x) = 3x + 2 and solve for y. Here, f-1(x) = (x - 2) / 3.
Example 2: Apply the process of switching x and y and solve algebraically. g-1(x) = x2 + 4.
Example 3: We can solve this equation by expressing it with an exponential function and finding its inverse. f-1(x) = 3x - 2.

Properties of Inverse Functions

Some important properties of inverse functions include:

The composition of a function and its inverse is equivalent to the identity function.
If f(x) has an inverse $\mathrm{f^{-1}(\mathrm{x})}$ , then the inverse of the inverse is the original function, i.e., $\mathrm{\left(f^{-1}\right)^{-1}(x)=f(x)}$
If f(x) is an increasing or decreasing function, its inverse $\mathrm{f^{-1}(x)}$ is also an increasing or decreasing function, respectively.

Tips and Tricks

Here are a few tips and tricks to keep in mind while dealing with inverse functions:

Always understand the domain and range of the given function to determine the domain and range of the inverse function.
To find the inverse of a function, interchange x and y, and solve for y algebraically.
Use the composition of functions property to verify the obtained inverse.
Sometimes, simplifying the function algebraically or applying algebraic rules can help find the inverse function quickly.
Practice solving problems from previous year management entrance exams to strengthen your understanding of the concept.