Which of the following function pairs are inverses?
A. [tex]$f(x)=2 x-3$[/tex] and [tex]$g(x)=\frac{x+3}{2}$[/tex]
B. [tex]$f(x)=-4 x$[/tex] and [tex]$g(x)=\frac{1}{4} x$[/tex]
C. [tex]$f(x)=7 x+1$[/tex] and [tex]$g(x)=7 x-1$[/tex]
D. [tex]$f(x)=x^2-9$[/tex] and [tex]$g(x)=x+3$[/tex]
Answer (1)
To determine if two functions are inverses, we verify if f ( g ( x )) = x and g ( f ( x )) = x .
For f ( x ) = 2 x − 3 and g ( x ) = 2 x + 3 , we find f ( g ( x )) = x and g ( f ( x )) = x .
For the other pairs, the condition f ( g ( x )) = x and g ( f ( x )) = x is not met.
Therefore, the inverse functions are: f ( x ) = 2 x − 3 and g ( x ) = 2 x + 3 .
Explanation
Understanding Inverse Functions We are given four pairs of functions and asked to identify which pair are inverses of each other. Two functions, f ( x ) and g ( x ) , are inverses if and only if f ( g ( x )) = x and g ( f ( x )) = x . We will check each pair to see if they satisfy this condition.
Analyzing Pair 1 Let's analyze the first pair: f ( x ) = 2 x − 3 and g ( x ) = 2 x + 3 .
We need to compute f ( g ( x )) and g ( f ( x )) .
f ( g ( x )) = f ( 2 x + 3 ) = 2 ( 2 x + 3 ) − 3 = ( x + 3 ) − 3 = x g ( f ( x )) = g ( 2 x − 3 ) = 2 ( 2 x − 3 ) + 3 = 2 2 x = x Since both f ( g ( x )) = x and g ( f ( x )) = x , the first pair of functions are inverses.
Analyzing Pair 2 Now let's analyze the second pair: f ( x ) = − 4 x and g ( x ) = 4 1 x .
We need to compute f ( g ( x )) and g ( f ( x )) .
f ( g ( x )) = f ( 4 1 x ) = − 4 ( 4 1 x ) = − x g ( f ( x )) = g ( − 4 x ) = 4 1 ( − 4 x ) = − x Since f ( g ( x )) = − x and g ( f ( x )) = − x , this pair of functions are NOT inverses.
Analyzing Pair 3 Now let's analyze the third pair: f ( x ) = 7 x + 1 and g ( x ) = 7 x − 1 .
We need to compute f ( g ( x )) and g ( f ( x )) .
f ( g ( x )) = f ( 7 x − 1 ) = 7 ( 7 x − 1 ) + 1 = 49 x − 7 + 1 = 49 x − 6 g ( f ( x )) = g ( 7 x + 1 ) = 7 ( 7 x + 1 ) − 1 = 49 x + 7 − 1 = 49 x + 6 Since f ( g ( x )) = 49 x − 6 and g ( f ( x )) = 49 x + 6 , this pair of functions are NOT inverses.
Analyzing Pair 4 Now let's analyze the fourth pair: f ( x ) = x 2 − 9 and g ( x ) = x + 3 .
We need to compute f ( g ( x )) and g ( f ( x )) .
f ( g ( x )) = f ( x + 3 ) = ( x + 3 ) 2 − 9 = x 2 + 6 x + 9 − 9 = x 2 + 6 x g ( f ( x )) = g ( x 2 − 9 ) = ( x 2 − 9 ) + 3 = x 2 − 6 Since f ( g ( x )) = x 2 + 6 x and g ( f ( x )) = x 2 − 6 , this pair of functions are NOT inverses.
Conclusion Therefore, the only pair of functions that are inverses of each other is f ( x ) = 2 x − 3 and g ( x ) = 2 x + 3 .
Examples
Inverse functions are useful in many real-world applications. For example, if f ( x ) converts temperature from Celsius to Fahrenheit, then its inverse g ( x ) converts temperature from Fahrenheit to Celsius. If f ( x ) = 5 9 x + 32 , then g ( x ) = 9 5 ( x − 32 ) . Another example is encoding and decoding messages. If f ( x ) is an encoding function, then g ( x ) is the decoding function that recovers the original message.
