Inverse Functions $$ f(2x^2 + x) = 7x - 3 $$

Inverse Functions

Inverse functions help us understand how inputs and outputs of a function can be reversed. In this lesson, we will analyze an inverse function problem step by step and find the value of the inverse function for a given input.

1. Theory of Inverse Functions

An inverse function reverses the effect of the original function. This means that the input and output values are swapped. If a function is written as f(x), its inverse is written as f^-1(x). Graphically, a function and its inverse are reflections of each other across the line y = x.

2. Concept Explanation

When a function is defined using an expression such as f(2x² + x) = 7x - 3, we can find information about its inverse by switching the roles of the input and output. This allows us to express the inverse function in terms of the original variables.

3. Steps or Formula

1. Swap the input and output to represent the inverse relationship.
2. Identify the expression inside the inverse function.
3. Solve for the value that makes the inverse input equal to the given number.
4. Substitute this value back into the inverse expression.

4. Example Problem

Given:

$$ f(2x^2 + x) = 7x - 3 $$

Question:

$$ f^{-1}(4) = ? $$

Step 1: Write the inverse relationship.

Since inverse functions swap inputs and outputs:

$$ f^{-1}(7x - 3) = 2x^2 + x $$

Step 2: Find the value of x that makes the input equal to 4.

$$ 7x - 3 = 4 $$ $$ 7x = 7 $$ $$ x = 1 $$

Step 3: Substitute x into the inverse expression.

$$ f^{-1}(4) = 2(1)^2 + 1 $$ $$ f^{-1}(4) = 2 + 1 = 3 $$

5. Graph Interpretation

The function y = 7x - 3 and its inverse y = \frac{x + 3}{7} are mirror images of each other across the line y = x. This reflection confirms the relationship between a function and its inverse.

Final Answer:

$$ f^{-1}(4) = 3 $$