Question

a composite function can be written as w(u(x)), where u and w are basic functions.
is ( f(x) = e^{x - 7} ) a composite function? if so, what are the \inner\ and \outer\ functions?
choose 1 answer:
a ( f ) is composite. the \inner\ function is ( e^x ) and the \outer\ function is ( x - 7 ).
b ( f ) is composite. the \inner\ function is ( x - 7 ) and the \outer\ function is ( e^x ).
c ( f ) is not a composite function.

Explanation:

Step1: Recall composite function definition

A composite function \( w(u(x)) \) means we first apply \( u \) to \( x \), then apply \( w \) to \( u(x) \). For \( f(x) = e^{x - 7} \), we need to identify two basic functions where one is inside the other.

Step2: Analyze the structure of \( f(x) \)

The function \( f(x) \) has the form of an exponential function where the exponent is \( x - 7 \). So, if we let the inner function \( u(x)=x - 7 \) (this is the function we apply first to \( x \)), and the outer function \( w(x)=e^{x} \) (this is the function we apply to \( u(x) \)), then \( w(u(x)) = e^{u(x)}=e^{x - 7} \), which matches \( f(x) \). Option A swaps inner and outer, and option C is incorrect as \( f(x) \) is composite.

Answer:

B. \( f \) is composite. The "inner" function is \( x - 7 \) and the "outer" function is \( e^{x} \).