Question

1.2 homework ~ graphing functions, composition of functions, and piece-w
score: 40/80 answered: 5/8
question 6
given the function $f(x) = 2x - 4$ and the function $g(x) = 6x^2 + 7x + 2$ determine
each of the following.
give your answer as an integer or a simplified fraction.
evaluate $f(g(6))$ $f(g(6)) = $
evaluate $g(f(7))$ $g(f(7)) = $
evaluate $f(g(3))$ $f(g(3)) = $
evaluate $f(f(2))$ $f(f(2)) = $
question help: video

Explanation:

Part 1: Evaluate \( f(g(6)) \)

Step 1: Find \( g(6) \)

Substitute \( x = 6 \) into \( g(x)=6x^{2}+7x + 2 \).
\( g(6)=6\times6^{2}+7\times6 + 2=6\times36+42 + 2=216+42 + 2=260 \)

Step 2: Find \( f(g(6))=f(260) \)

Substitute \( x = 260 \) into \( f(x)=2x - 4 \).
\( f(260)=2\times260-4 = 520 - 4=516 \)

Step 1: Find \( f(7) \)

Substitute \( x = 7 \) into \( f(x)=2x - 4 \).
\( f(7)=2\times7-4 = 14 - 4 = 10 \)

Step 2: Find \( g(f(7))=g(10) \)

Substitute \( x = 10 \) into \( g(x)=6x^{2}+7x + 2 \).
\( g(10)=6\times10^{2}+7\times10+2=6\times100 + 70+2=600+70 + 2=672 \)

Step 1: Find \( g(3) \)

Substitute \( x = 3 \) into \( g(x)=6x^{2}+7x + 2 \).
\( g(3)=6\times3^{2}+7\times3+2=6\times9+21 + 2=54+21 + 2=77 \)

Step 2: Find \( f(g(3))=f(77) \)

Substitute \( x = 77 \) into \( f(x)=2x - 4 \).
\( f(77)=2\times77-4=154 - 4 = 150 \)

Answer:

\( f(g(6)) = 516 \)

Part 2: Evaluate \( g(f(7)) \)