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 \( g(x) = \frac{1}{2}(x - 3) \).
Answer for screen readers
The functions ( f(x) = 2x + 3 ) and ( g(x) = \frac{1}{2}(x - 3) ) are inverses of each other.
Steps to Solve
- Identify the Functions
Letâ€™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.