Question

Consider the following functions:

f(x) = 7x
g(x) = x
h(x) = x² + 7
Givethe functions below with their compositions:
A. (gºf)(x)
B. (hºg)(x)
C. (fºg)(x)
D. (gºh)(x)
E. (hºf)(x)
F. (fºh)(x)

Answer 1

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.

Answer 2

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.