Question

Let h(x)=(g o f)(x)=√x+3
Which of the following could be a possible decomposition of h(x)?

Answer 1

To decompose h(x) = √(x + 3), one can select f(x) = x + 3 and g(x) = √x, or f(x) = x and g(x) = √(x + 3), so that (g o f)(x) correctly yields h(x).

Step-by-step explanation:

The question asks to determine a possible decomposition of the composite function h(x) = √(x + 3), given that h(x) is the composition of two functions (g o f)(x). To find possible functions f(x) and g(x) that compose h(x), we can consider various combinations of basic functions such as linear functions, quadratic functions, and square root functions.

One possible decomposition could be:

• f(x) = x + 3
• g(x) = √x

This way, when we apply g to the result of f(x), g(f(x)) = g(x + 3) = √(x + 3), which matches our given h(x).

Another possible decomposition could be:

• f(x) = x
• g(x) = √(x + 3)

Again, the composition g(f(x)) would yield the correct function h(x).

Answer 2

In this case, h(x) = sqrt(x) + 3

A. f(x)=x+3; g(x)=√x
B. f(x)=x; g(x)=x+3
C. f(x)=√x; g(x)=x+3
D. f(x)=3x; g(x)=√x

Again, you need to find a function f(x) that once evaluated in g(x) gives us h(x)

h(x) = g(f(x))

Looking at the options, the answer is C.

g(f(x)) = f(x) + 3 = sqrt (x) + 3 = h(x)