Inverse Functions Practice Questions

An inverse function undoes the actions of another function, effectively reversing its operation. It swaps input and output values, allowing for the retrieval of the original input from a given output. This article will teach us how to solve the questions based on the inverse function.

The inverse of a function f is a function f⁻¹ such that:

f{f ⁻¹ (y)} = y for all y ∈ Range (f)

and

f⁻¹{f(x)} = x for all x ∈ 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) = e^x 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 of some common function are added in the table below:

Q1. Find the inverse of the function ?(?)=2?+3

Solution:

Replace f(x) with y:

y = 2x + 3

Swap x and y: x = 2y + 3

Solve for y: y = (x-3)/2​

So, f⁻¹(x) = (x-3)/2​

Q2. Determine the inverse of ?(?) = (?-4)/3

Solution:

Replace f(x) with y:

y = 3x−4​

Swap x and y: x = (y-4)/3

Solve for y: y = 3x + 4

So, f^-1(x) = 3x + 4

Q3. If f(x) = x² for x ≥ 0, find f^-1(x):

Solution:

Replace f(x) with y:

y = x²

Swap x and y: x = y²

Solve for y: y = √x​

So, f^-1(x) = √x​ for x ≥ 0

Q4. What is the inverse function of f(x)=tan⁡(x) -?/2<x<?/2.

Solution:

Replace f(x) with y:

y = tan(x)

Swap x and y: x = tan(y)

Solve for y: y = arctan(x)

So, f⁻¹(x) = arctan(x)

Q5. If f(x)=x 1/x​ for x≠0, find f^-1(x)

Solution:

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/x​

Q6. What is the inverse function of f(x) = x³

Solution:

Replace f(x) with y:

y = x³

Swap x and y: x = y³

Solve for y: y = ∛x​

So, f^-1(x) = ∛x​

Q7. Determine f^-1(x) for the function f(x)=e^x

Solution:

Replace f(x) with y:

y = e^x

Swap x and y: x = e^y

Solve for y: y = ln(x)

So, f^-1(x) = ln(x)

Q1. Find the inverse of the function f(x) = (x+2)/(3x-1)​.

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 −1 ≤ x ≤ 1, find f^-1(x).

Q5. What is the inverse function of f(x) = arctan(x)?

What is an Inverse Function?

An inverse function is a function that reverses the operations of a given function. If you have a function f that maps an element x to an element y, then the inverse function f⁻¹ maps the element y back to the element x. In other words, if f(x) = y, then f⁻¹(y) = x.

What are 4 rules to Find Inverse of a Function?

4 rules to find the inverse of a function are:

• Replace f(x) with y
• Swap x and y
• Solve the equation for y
• Replace y with f^-1(x)

What are 3 Steps of Solving an Inverse Function?

3 steps to solving an inverse function are:

• Replace f(x) with y
• Swap x and y
• Solve the equation for y

What is Inverse of x³?

Inverse of x³ is ∛x​.

What is Inverse of 1?

Inverse of 1 is 1, because any number raised to the power of 0 is 1.