Question

Find (f(x)) and (g(x)) so that the function can be described as (y = f(g(x))). (5 points) (y = 2/x² + 9). Please show all steps.

a) (f(x) = 2) and (g(x) = x² + 9)

b) (f(x) = 2/x) and (g(x) = x² + 9)

c) (f(x) = 2/x² + 9) and (g(x) = 1)

d) (f(x) = 2x) and (g(x) = 1/x² + 9)

Answer 1

The correct functions f(x) and g(x) that compose y = 2/x² + 9 are f(x) = 2/x and g(x) = x² + 9, which corresponds to option b. By composing f(g(x)), we get the original function, confirming our solution.

Step-by-step explanation:

Function Composition

To find functions f(x) and g(x) such that y = f(g(x)), for the function y = 2/x² + 9, we need to identify an inner function (g(x)) and an outer function (f(x)) that can be composed to form the given function. Looking at the options provided, we can see that option b creates the correct composition:

• If we let g(x) = x² + 9, then g(x) incorporates the variable x in a quadratic function with a constant added. This appears to be part of the given function.
• Next, we let f(x) = 2/x, which is a transformation of x that inversely relates to x and multiplies it by 2. This also is a part of the original function.

When composing these two functions:

1. First, we compute g(x) which gives us x² + 9.
2. Then, we apply f(x) to the result of g(x) which produces f(g(x)) = 2/(x² + 9).

This composition matches the original function, leading to the conclusion that the correct answer is option b: (f(x) = 2/x) and (g(x) = x² + 9).