Question

for the follwoing function, find functions f(x) and g(x) so that given function can be expressed as h(x)=f(g(x)).
h(x)=(x+2)^2

Answer 1

To express the function \( h(x) = (x + 2)^2 \) in the form \( h(x) = f(g(x)) \), we need to identify two functions \( f(x) \) and \( g(x) \) such that when we substitute \( g(x) \) into \( f(x) \), we retrieve \( h(x) \).

A straightforward choice for \( g(x) \) would be \( g(x) = x + 2 \). This simplifies the expression inside the function.

Next, we want to define \( f(x) \). Since \( h(x) \) can be seen as squaring the result of \( g(x) \), we can choose \( f(x) = x^2 \).

In this case, we have:
- \( g(x) = x + 2 \)
- \( f(x) = x^2 \)

Now, we can verify that \( h(x) = f(g(x)) \):
\[
h(x) = f(g(x)) = f(x + 2) = (x + 2)^2
\]

Thus, we have successfully expressed \( h(x) \) in the desired form:
\[
h(x) = f(g(x))
\]

The final functions are:
- \( f(x) = x^2 \)
- \( g(x) = x + 2 \)