Final answer:
The pair that will create the composite function G(F(x)) = (x - 5)^3 + x - 5 is Option D: F(x) = x - 5 and G(x) = x^3 + x, as it matches the form of the given composite function when F(x) is substituted into G(x).
Step-by-step explanation:
The question asks which pair of functions F(x) and G(x) will create the composite function G(F(x)) = (x - 5)^3 + x - 5 when composed.
To find G(F(x)), we need to substitute the output of F(x) into G(x). Looking at the options provided and the form of G(F(x)), the correct pair must result in a composition where the (x - 5) term is cubed, and an additional (x - 5) term is added.
Option A: F(x) = x - 5 and G(x) = (x - 5)^3. Here, plugging F(x) into G(x) gives G(F(x)) = ((x - 5) - 5)^3, which does not match the desired form, so this is not the correct option.
Option B: F(x) = (x - 5)^3 and G(x) = x - 5. Plugging F(x) into G(x) gives G(F(x)) = ((x - 5)^3) - 5, which again is not the correct form, so this is also incorrect.
Option C: F(x) = x^2 + x - 5 and G(x) = x - 5. Substituting F(x) into G(x) would not produce the desired form since you would end up subtracting 5 from a polynomial that is not simply (x - 5).
Option D: F(x) = x - 5 and G(x) = x^3 + x. If we substitute F(x) into G(x), we get G(F(x)) = (x - 5)^3 + (x - 5), which exactly matches the desired composite function, making this the correct answer.