INVERSE FUNCTIONS

Inverse Functions

Definition Of Inverse Functions

f ^-1(x) is said to be the inverse of f(x)
if f(f ^-1(x)) = f ^-1(f(x)) = x. It is obtained by interchanging the variables x and y of the function.

More About Inverse Functions

Inverse of a function is the reflection of the function about the line y = x.
Inverse of a function does not necessarily always be a function.

Video Examples: Inverse Functions - The Basics

Example of Inverse Functions

y = 3x - 5 is a function where x = 3, 4, 5. y = 3x - 5 = 4, 7, and 10 by substituting the values of x.
Thus, the function is {(3, 4), (4, 7), (5, 10)}.
The inverse of the function is {(4, 3), (7, 4) (10, 5)} by interchanging the first and second co-ordinates.

Solved Example on Inverse Functions

Ques: Find the inverse of the function y = 4x - 7/4.

Choices:

A. y = 4x + 3
B. y = 4/4x + 7
C. y = 4x +7/4
D. y = x + 7
Correct Answer: C

Solution:

Step 1: y = 4x - 7/4.
Step 2: Interchange x and y and find y in terms of x.
Step 3: x = 4y - 7/4 [Interchange x and y.]
Step 4: 4x = 4y - 7 [Multiply throughout by 4.]
Step 5: 4x + 7 = 4y [Add 7 to both sides of the equation.]
Step 6: 4x + 7/4 = y
Step 7: The inverse function is y = 4x + 7/4.

Quick Summary

Inverse of a function is the reflection of the function about the line y = x.
Inverse of a function does not necessarily always be a function.
To find the inverse, interchange x and y and solve for y.

\[ f(f^{-1}(x)) = f^{-1}(f(x)) = x \]

🍎 Teacher Insights

Emphasize the graphical representation of inverse functions as reflections across the line y=x. Use various examples, including those where the inverse is not a function, to illustrate the concept thoroughly.

🎓 Prerequisites

Functions
Algebra
Coordinate Geometry

Check Your Knowledge

Q1: Find the inverse of the function y = 2x + 3.

A. y = 2x - 3 B. y = (x - 3)/2 C. y = (x + 3)/2 D. y = -2x - 3

Q2: Which of the following statements is true about inverse functions?

A. The inverse of a function is always a function. B. The inverse of a function is a reflection about the x-axis. C. The inverse of a function is a reflection about the y-axis D. The inverse of a function is a reflection about the line y = x.

Frequently Asked Questions

Q: How do I find the inverse of a function?
A: Interchange x and y in the function's equation, then solve for y. The resulting equation, with y isolated, represents the inverse function.

Q: Is the inverse always a function?
A: No, the inverse is not always a function. It must pass the vertical line test to be considered a function.