How to tell if two functions are inverses?

Understand the Problem

The question is asking how to determine if two given functions are inverses of each other. This typically involves checking if the composition of the two functions yields the identity function.

Answer

The functions are inverses: \( f(x) = 2x + 3 \) and \( g(x) = \frac{1}{2}(x - 3) \).

Steps to Solve

Identify the Functions

Let’s define the two functions as ( f(x) ) and ( g(x) ). For example, suppose ( f(x) = 2x + 3 ) and ( g(x) = \frac{1}{2}(x - 3) ).

Compose the Functions

We need to find ( f(g(x)) ) and ( g(f(x)) ) to see if they equal ( x ).

First, calculate ( f(g(x)) ):

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

Simplifying this, we get:

[ f(g(x)) = (x - 3) + 3 = x ]

Check the Other Composition

Now calculate ( g(f(x)) ):

[ g(f(x)) = g(2x + 3) = \frac{1}{2}((2x + 3) - 3) ]

Simplifying this, we find:

[ g(f(x)) = \frac{1}{2}(2x) = x ]

Determine If They Are Inverses

Since both compositions yield ( x ) (i.e., ( f(g(x)) = x ) and ( g(f(x)) = x )), we confirm that the functions ( f(x) ) and ( g(x) ) are indeed inverses of each other.

The functions ( f(x) = 2x + 3 ) and ( g(x) = \frac{1}{2}(x - 3) ) are inverses of each other.

More Information

When two functions are inverses, the output of one function becomes the input of the other and vice versa, resulting in the identity function. This property is crucial in calculus and algebra, especially in solving equations.

Tips

Forgetting to check both compositions. To confirm that two functions are inverses, both ( f(g(x)) ) and ( g(f(x)) ) must equal ( x ).
Simplifying incorrectly. Pay close attention to each step to ensure that each simplification is accurate.

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