Final Answer:
A. (gºf)(x) = x
B. (hºg)(x) = x² + 7
C. (fºg)(x) = 7x
D. (gºh)(x) = x² + 7
E. (hºf)(x) = 49x² + 7
F. (fºh)(x) = 7x² + 7
Step-by-step explanation:
In function composition, (gºf)(x) means "g of f of x." Let's break down each composition:
A. (gºf)(x):
(gºf)(x) = g(f(x)) = g(7x) = 7x. The function g simply returns the input, so (gºf)(x) = x.
B. (hºg)(x):
(hºg)(x) = h(g(x)) = h(x) = x² + 7. The function h squares the input and adds 7.
C. (fºg)(x):
(fºg)(x) = f(g(x)) = f(x) = 7x. The function f multiplies the input by 7.
D. (gºh)(x):
(gºh)(x) = g(h(x)) = g(x² + 7) = x² + 7. The function g is the identity function, so it returns the input.
E. (hºf)(x):
(hºf)(x) = h(f(x)) = h(7x) = (7x)² + 7 = 49x² + 7. The function h squares the input and adds 7.
F. (fºh)(x):
(fºh)(x) = f(h(x)) = f(x² + 7) = 7(x² + 7). Distributing, we get 7x² + 49.
In summary, the compositions result in the given functions. These answers are obtained by substituting the expressions of each function into the corresponding composition and simplifying the expressions.