Question

(c) \\( f(g(-1)) \\)

if \\( f(x) = 3x + 1 \\) and \\( g(x) = x^2 - 4 \\), determine each of the following

a) \\( f(0) \times g(0) \\)
\\( 1 \times -4 = -4 \\)
b) \\( f(-2) + g(-3) \\)
c) \\( f(g(2)) \\)
d) \\( g(f(2)) \\)

Explanation:

Part (b)

Step1: Calculate \( f(-2) \)

Substitute \( x = -2 \) into \( f(x) = 3x + 1 \):
\( f(-2) = 3(-2) + 1 = -6 + 1 = -5 \)

Step2: Calculate \( g(-3) \)

Substitute \( x = -3 \) into \( g(x) = x^2 - 4 \):
\( g(-3) = (-3)^2 - 4 = 9 - 4 = 5 \)

Step3: Add the results

\( f(-2) + g(-3) = -5 + 5 = 0 \)

Step1: Calculate \( g(2) \)

Substitute \( x = 2 \) into \( g(x) = x^2 - 4 \):
\( g(2) = (2)^2 - 4 = 4 - 4 = 0 \)

Step2: Calculate \( f(g(2)) = f(0) \)

Substitute \( x = 0 \) into \( f(x) = 3x + 1 \):
\( f(0) = 3(0) + 1 = 0 + 1 = 1 \)

Step1: Calculate \( f(2) \)

Substitute \( x = 2 \) into \( f(x) = 3x + 1 \):
\( f(2) = 3(2) + 1 = 6 + 1 = 7 \)

Step2: Calculate \( g(f(2)) = g(7) \)

Substitute \( x = 7 \) into \( g(x) = x^2 - 4 \):
\( g(7) = (7)^2 - 4 = 49 - 4 = 45 \)

Answer:

\( 0 \)

Part (c)