Inverse Of A Function Examples With Solutions

Inverse of a Function: Examples with Solutions

Understanding the inverse of a function is a fundamental concept in algebra and calculus, crucial for solving equations, analyzing relationships between variables, and modeling real-world scenarios. Even so, an inverse function essentially reverses the operation of the original function, allowing us to "undo" its effects. This article explores the definition of inverse functions, provides step-by-step examples with detailed solutions, and explains the mathematical principles behind their existence and properties.

What Is an Inverse Function?

Given a function ( f(x) ), its inverse function, denoted as ( f^{-1}(x) ), is defined such that:

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

In simpler terms, if applying ( f ) and then ( f^{-1} ) (or vice versa) to a value returns the original input, the two functions are inverses of each other. For an inverse function to exist, the original function must be one-to-one, meaning each output corresponds to exactly one input Less friction, more output..

Steps to Find the Inverse of a Function

To find the inverse of a function ( f(x) ), follow these steps:

1. Replace ( f(x) ) with ( y ).
2. Swap ( x ) and ( y ).
3. Solve for ( y ).
4. Replace ( y ) with ( f^{-1}(x) ).

Let’s apply these steps to several examples Surprisingly effective..

Example 1: Linear Function

Problem: Find the inverse of ( f(x) = 2x + 3 ).

Solution:

1. Replace ( f(x) ) with ( y ): [ y = 2x + 3 ]

2. Swap ( x ) and ( y ): [ x = 2y + 3 ]

3. Solve for ( y ): [ x - 3 = 2y \quad \Rightarrow \quad y = \frac{x - 3}{2} ]

4. Replace ( y ) with ( f^{-1}(x) ): [ f^{-1}(x) = \frac{x - 3}{2} ]

Verification: [ f(f^{-1}(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x ] [ f^{-1}(f(x)) = f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x ]

The functions satisfy the inverse property.

Example 2: Quadratic Function

Problem: Find the inverse of ( f(x) = x^2 ), and explain why it might not exist for all real numbers.

Solution:

1. Replace ( f(x) ) with ( y ): [ y = x^2 ]

2. Swap ( x ) and ( y ): [ x = y^2 ]

3. Solve for ( y ): [ y = \pm \sqrt{x} ]

Here, we encounter a problem: the quadratic function ( f(x) = x^2 ) is not one-to-one over all real numbers because it fails the horizontal line test (a horizontal line intersects the parabola twice). To define an inverse, we must restrict the domain of ( f(x) ) Simple, but easy to overlook..

• If we restrict ( f(x) ) to ( x \geq 0 ), the inverse is ( f^{-1}(x) = \sqrt{x} ).
• If we restrict ( f(x) ) to ( x \leq 0 ), the inverse is ( f^{-1}(x) = -\sqrt{x} ).

Conclusion: The inverse of ( f(x) = x^2 ) exists only when we restrict its domain It's one of those things that adds up. Simple as that..

Example 3: Rational Function

Problem: Find the inverse of ( f(x) = \frac{1}{x} ), where ( x \neq 0 ).

Solution:

1. Replace ( f(x) ) with ( y ): [ y = \frac{1}{x} ]

2. Swap ( x ) and ( y ): [ x = \frac{1}{y} ]

3. Solve for ( y ): [ y = \frac{1}{x} ]

4. Replace ( y ) with ( f^{-1}(x) ): [ f^{-1}(x) = \frac{1}{x} ]

Verification: [ f(f^{-1}(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)} = x ] [ f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)} = x ]

Interestingly, ( f(x) = \frac{1}{x} ) is its own inverse.

Example 4: Cube Root Function

Problem: Find the inverse of ( f(x) = \sqrt[3]{x} ).

Solution:

1. Replace ( f(x) ) with ( y ): [ y = \sqrt[3]{x} ]

2. Swap ( x ) and ( y ): [