Inverse functions are those functions which reverse the actions of another function and reversing its operation. It swaps input and output values, allowing for the retrieval of the original input from a given output.
This article consist of a series of Inverse Functions Practice Questions focused at strengthening your understanding and proficiency in this fundamental concept of Inverse functions.
Table of Content
What are Inverse functions
The inverse function is inverse of a function.
let f be a function then fβ1 represents the inverse of function (f), such that:
f{f β1 (y)} = y for all y β Range (f)
and
fβ1{f(x)} = x forΒ allΒ x β Domain(f)
- Graph of an inverse function is a reflection of the original function across the line y = x.
If f(x) = log x then its inverse is, f-1(x) = ex and the graph for the same is added below:
- To graph the inverse function, interchange the roles of x and y. If the original graph passes through (a, b), the inverse graph will pass through (b, a).
Inverse function of Some Common Functions
Inverse function of some common functions are added in the table below:
Function | Inverse Function | Domain of Function | Range of Inverse Function |
|---|---|---|---|
f(x) = x+a | fβ1(x) = xβa | All real numbers | All real numbers |
f(x) = x-a | fβ1(x) = x+a | All real numbers | All real numbers |
f(x) = ax (aβ 0) | fβ1(x) = x/a | All real numbers | All real numbers |
f(x) = x/a (aβ 0) | fβ1(x) = ax | All real numbers | All real numbers |
f(x) = x2 | fβ1(x) = βxβ | x β₯ 0 | x β₯ 0 |
f(x) = βxβ | fβ1(x) = x2 | x β₯ 0 | x β₯ 0 |
f(x) = x3 | fβ1(x) = βxβ | All real numbers | All real numbers |
f(x) = βx | fβ1(x) = x3 | All real numbers | All real numbers |
f(x) = ln(x) | fβ1(x) = ex | x > 0 | All real numbers |
f(x) = ex | fβ1(x) = ln(x) | All real numbers | x > 0 |
f(x) = sin(x) | fβ1(x) = arcsin(x) | βΟβ/2 β€ x β€ Ο/2β | β1 β€ x β€ 1 |
f(x) = cos(x) | fβ1(x) = arccos(x) | 0 β€ x β€ Ο | β1β€ x β€ 1 |
f(x) = tan(x) | fβ1(x) = arctan(x) | βΟβ/2 < x < Ο/2β | All real numbers |
Inverse Functions Practice Questions - Solved
1. Find the inverse of the function ?(?)=2?+3
Replace f(x) with y:
y = 2x + 3
Swap x and y: x = 2y + 3
Solve for y: y = (x-3)/2β
So, fβ1(x) = (x-3)/2β
2. Determine the inverse of ?(?) = (?-4)/3
Replace f(x) with y:
y = 3xβ4β
Swap x and y: x = (y-4)/3
Solve for y: y = 3x + 4
So, f-1(x) = 3x + 4
3. If f(x) = x2 for x β₯ 0, find f-1(x):
Replace f(x) with y:
y = x2
Swap x and y: x = y2
Solve for y: y = βxβ
So, f-1(x) = βxβ for x β₯ 0
4. What is the inverse function of f(x)=tanβ‘(x) -?/2<x<?/2.
Replace f(x) with y:
y = tan(x)
Swap x and y: x = tan(y)
Solve for y: y = arctan(x)
So, fβ1(x) = arctan(x)
5. If f(x)=x 1/xβ for xβ 0, find f-1(x)
Replace f(x) with y:
y = 1/x
Swap x and y: x=1/y
Solve for y: y= 1/x
So, f-1(x)= 1/xβ
6. What is the inverse function of f(x) = x3
Replace f(x) with y:
y = x3
Swap x and y: x = y3
Solve for y: y = βxβ
So, f-1(x) = βxβ
7. Determine f-1(x) for the function f(x)=ex
Replace f(x) with y:
y = ex
Swap x and y: x = ey
Solve for y: y = ln(x)
So, f-1(x) = ln(x)
Inverse Functions Practice Questions - Unsolved
Q1. Find the inverse of the function f(x) = (x+2)/(3x-1)β.
Q2. Determine f-1(x) for the function f(x) = log(x).
Q3. If f(x) = arcsin (x), what is f-1(x)?
Q4. Given f(x) = arccos (x) for β1 β€ x β€ 1, find f-1(x).
Q5. What is the inverse function of f(x) = arctan(x)?
