Question

Given f(x) = 2x, g(x) = x + 3, h(x) = 2x + 6, show every step to find and simplify the function rules. Note any values that must be excluded from the domain.

a. (f.g)(x)
b. (g.h)(x)
c. (f:h)(x), h(x) not equal to 0
d. (h:g)(x), g(x) not equal to 0
e. (g:f)(x)

Answer 1

To find the function rules, the given functions are substituted into each other to create composite functions. The composite functions are simplified by performing the operations. The excluded values from the domain are where the denominators of the functions equal zero.

Step-by-step explanation:

1. To find (f.g)(x), we substitute g(x) into f(x). So, (f.g)(x) = f(g(x)) = f(x + 3) = 2(x + 3).
2. To find (g.h)(x), we substitute h(x) into g(x). So, (g.h)(x) = g(h(x)) = g(2x + 6) = 2x + 6 + 3 = 2x + 9.
3. To find (f:h)(x), we divide f(x) by h(x) by substituting their respective function rules. So, (f:h)(x) = f(x) / h(x) = (2x) / (2x + 6).
4. To find (h:g)(x), we divide h(x) by g(x) by substituting their respective function rules. So, (h:g)(x) = h(x) / g(x) = (2x + 6) / (x + 3).
5. To find (g:f)(x), we substitute f(x) into g(x). So, (g:f)(x) = g(f(x)) = g(2x) = 2x + 3.