8) From the functions listed on the right, find two whose composite function is.
a)
2
h x x 1
b)
h x x 3
c)
2
h x x x 6 9
d)
h x x 2
a) The function h(x) can be written as h(x) = x^2 + 1.
b) The function h(x) can be written as h(x) = -3.
c) The function h(x) can be written as h(x) = x^2 - x + 6.
d) The function h(x) can be written as h(x) = -2.
To find two functions whose composite function is h(x) for each case:
a) If we let g(x) = x + 1 and f(x) = x^2, then h(g(x)) = h(x + 1) = (x + 1)^2 + 1 = x^2 + 2x + 2. Therefore, g(x) = x + 1 and f(x) = x^2 are two functions whose composite function is h(x) = x^2 + 1.
b) Since h(x) = -3 for all values of x, any function g(x) will result in h(g(x)) = -3.
c) If we let g(x) = x + 2 and f(x) = x^2 - x + 4, then h(g(x)) = h(x + 2) = (x + 2)^2 - (x + 2) + 4 = x^2 + 2x + 4 - x - 2 + 4 = x^2 + x + 6. Therefore, g(x) = x + 2 and f(x) = x^2 - x + 4 are two functions whose composite function is h(x) = x^2 - x + 6.
d) Since h(x) = -2 for all values of x, any function g(x) will result in h(g(x)) = -2.
To find two functions whose composite function matches the given function, we need to observe the format of the composite function.
a) The composite function is h(x) = x^2 + 1.
The function x^2 matches the format of h(x) = x^2, and the function f(x) = x + 1 matches the format of h(x) = x + 1. Therefore, we can select f(x) = x + 1 and g(x) = x^2 as the two functions whose composite function matches h(x).
b) The composite function is h(x) = -3.
To match this function, we can select f(x) = -3 and g(x) = x^2. Therefore, f(x) = -3 and g(x) = x^2 are the functions whose composite function matches h(x).
c) The composite function is h(x) = x^2 - x + 6 - 9.
To match this function, we can select f(x) = x^2 + 6 and g(x) = x - 9. Therefore, f(x) = x^2 + 6 and g(x) = x - 9 are the functions whose composite function matches h(x).
d) The composite function is h(x) = -2.
To match this function, we can select f(x) = -2 and g(x) = x. Therefore, f(x) = -2 and g(x) = x are the functions whose composite function matches h(x).