Find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.
$f(x)=3 x+6 \text { and } g(x)=\frac{x-6}{3}$
a. $f(g(x))=\frac{3}{3}$ (Simplify your answer.)
b. $g(f(x))=x$ (Simplify your answer.)
c.
$f$ and $g$ are inverses of each other.
$f$ and $g$ are not inverses of each other.
Answer (2)
Calculate f ( g ( x )) by substituting g ( x ) into f ( x ) and simplify: f ( g ( x )) = 3 ( 3 x − 6 ) + 6 = x .
Calculate g ( f ( x )) by substituting f ( x ) into g ( x ) and simplify: g ( f ( x )) = 3 ( 3 x + 6 ) − 6 = x .
Since f ( g ( x )) = x and g ( f ( x )) = x , the functions are inverses of each other.
Therefore, the final answer is: f and g are inverses of each other and f ( g ( x )) = x and g ( f ( x )) = x .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 x + 6 and g ( x ) = f r a c x − 6 3 , and we need to find f ( g ( x )) and g ( f ( x )) to determine if they are inverses of each other. Two functions are inverses if and only if f ( g ( x )) = x and g ( f ( x )) = x .
Calculating f(g(x)) First, let's find f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( 3 x − 6 ) = 3 ( 3 x − 6 ) + 6 Simplifying the expression: f ( g ( x )) = ( x − 6 ) + 6 = x So, f ( g ( x )) = x .
Calculating g(f(x)) Next, let's find g ( f ( x )) . We substitute f ( x ) into g ( x ) :
g ( f ( x )) = g ( 3 x + 6 ) = 3 ( 3 x + 6 ) − 6 Simplifying the expression: g ( f ( x )) = 3 3 x = x So, g ( f ( x )) = x .
Determining if f and g are Inverses Since f ( g ( x )) = x and g ( f ( x )) = x , the functions f ( x ) and g ( x ) are inverses of each other.
Final Answer a. f ( g ( x )) = x b. g ( f ( x )) = x c. f and g are inverses of each other.
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then g ( x ) would convert Fahrenheit back to Celsius. If f ( x ) = 5 9 x + 32 converts Celsius to Fahrenheit, then its inverse g ( x ) = 9 5 ( x − 32 ) converts Fahrenheit to Celsius. This concept is useful in many applications, such as cooking, weather forecasting, and scientific research.
We calculated f ( g ( x )) = x and g ( f ( x )) = x , confirming that the functions are inverses of each other. Therefore, the answer is that f and g are inverses of each other.
;
