Answer:
Explanation:
(a) To find fg(x), we need to substitute g(x) into f(x) and simplify:
f(x) = 3x + 2
g(x) = (x - 3)²
Substituting g(x) into f(x):
fg(x) = f(g(x))
= f((x - 3)²)
Now, let's simplify the expression (x - 3)²:
(x - 3)² = (x - 3)(x - 3)
= x² - 6x + 9
Substituting this back into fg(x):
fg(x) = f((x - 3)²)
= 3(x² - 6x + 9) + 2
= 3x² - 18x + 27 + 2
= 3x² - 18x + 29
Therefore, fg(x) = 3x² - 18x + 29.
(b) To find ¹(-12), we need to substitute x = -12 into the function f(x) and evaluate:
f(x) = 3x + 2
Substituting x = -12:
f(-12) = 3(-12) + 2
= -36 + 2
= -34
Therefore, ¹(-12) = -34.