For [tex]f(x)=3 x-9[/tex] and [tex]g(x)=\frac{x+9}{3}[/tex] find:
a. [tex](f \circ g)(x)[/tex];
b. [tex](g \circ f)(x)[/tex];
c. [tex](f \circ g)(9)[/tex]
Answer (1)
Find the composite function ( f c i rc g ) ( x ) by substituting g ( x ) into f ( x ) .
Simplify the expression: f ( g ( x )) = 3 ( f r a c x + 9 3 ) − 9 = x + 9 − 9 = x .
The composite function ( f c i rc g ) ( x ) simplifies to x .
The final answer is x .
Explanation
Finding the composite function (f \circ g)(x) We are given two functions, f ( x ) = 3 x − 9 and g ( x ) = f r a c x + 9 3 , and we need to find the composite function ( f c i rc g ) ( x ) , which means f ( g ( x )) . We will substitute g ( x ) into f ( x ) and simplify the expression.
Simplifying the expression To find ( f c i rc g ) ( x ) , we substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( f r a c x + 9 3 ) = 3 ( f r a c x + 9 3 ) − 9 Now, we simplify the expression: f ( g ( x )) = ( x + 9 ) − 9 = x So, ( f c i rc g ) ( x ) = x .
Final result for (f \circ g)(x) Therefore, ( f c i rc g ) ( x ) = x .
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that marks down all items by 10% and then applies a $5 off coupon. If f ( x ) = 0.9 x represents the 10% markdown and g ( x ) = x − 5 represents the $5 off coupon, then the composite function ( g c i rc f ) ( x ) = 0.9 x − 5 represents the final price after both the markdown and the coupon are applied. Understanding composite functions helps in analyzing such sequential operations.
