Final answer:
To find the composite functions (f◦g) and (g◦f), substitute one function into the other. The values of (f◦g)(x) and (g◦f)(x) were calculated as 10x² - 12 and 20x² - 40x + 15, respectively. When -1 is substituted into these composite functions, the values are -2 and 75, respectively.
Step-by-step explanation:
a. To find (f◦g)(x), we need to substitute g(x) into f(x). So, (f◦g)(x) = f(g(x)).
Substituting g(x) = 5x² - 5 into f(x) = 2x - 2, we get:
(f◦g)(x) = 2(g(x)) - 2 = 2(5x² - 5) - 2 = 10x² - 10 - 2 = 10x² - 12
b. To find (g◦f)(x), we need to substitute f(x) into g(x). So, (g◦f)(x) = g(f(x)).
Substituting f(x) = 2x - 2 into g(x) = 5x² - 5, we get:
(g◦f)(x) = 5(f(x))² - 5 = 5(2x - 2)² - 5 = 5(4x² - 8x + 4) - 5 = 20x² - 40x + 20 - 5 = 20x² - 40x + 15
c. To find (f◦g)(-1), we substitute -1 into (f◦g)(x) = 10x² - 12.
(f◦g)(-1) = 10(-1)² - 12 = 10(1) - 12 = 10 - 12 = -2
d. To find (g◦f)(-1), we substitute -1 into (g◦f)(x) = 20x² - 40x + 15.
(g◦f)(-1) = 20(-1)² - 40(-1) + 15 = 20(1) + 40 + 15 = 20 + 40 + 15 = 75