Question

category 24
1° medium
the function f satisfies f(3) = 5. the function g is defined by g(x) = \frac{2x^3 - 98}{19}. what is the value of g(f(3))?
2° hard
if the function f is defined by f(x) = 4x - 19, and the function g is defined by g(x) = 3f(x - 1) - 3, what is the value of g(7)?
a) 2
b) 5
c) 12
d) 15
3° hard
the function g is defined by f(x) = x^2 - x, and the function f is defined by g(x) = x^3 - 20. what is the value of f(g(3))?
a) 6
b) 14
c) 27
d) 42

Explanation:

1st Problem

Step1: Find \( f(3) \)

Given \( f(3) = 5 \).

Step2: Substitute into \( g(x) \)

Now, we need to find \( g(f(3)) = g(5) \). The function \( g(x)=\frac{2x^{3}-98}{19} \). Substitute \( x = 5 \) into \( g(x) \):

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Step1: Find \( f(x - 1) \) when \( x=7 \)

First, for \( g(7)=3f(7 - 1)-3=3f(6)-3 \). We need to find \( f(6) \) using \( f(x)=4x - 19 \). Substitute \( x = 6 \) into \( f(x) \):
\( f(6)=4\times6-19=24 - 19 = 5 \)

Step2: Substitute \( f(6) \) into \( g(7) \)

Now substitute \( f(6) = 5 \) into \( g(7) \):
\( g(7)=3\times5-3=15 - 3 = 12 \)

Step1: Find \( g(3) \)

The function \( g(x)=x^{3}-20 \). Substitute \( x = 3 \) into \( g(x) \):
\( g(3)=3^{3}-20=27 - 20 = 7 \)

Step2: Substitute \( g(3) \) into \( f(x) \)

The function \( f(x)=x^{2}-x \). Substitute \( x = 7 \) into \( f(x) \):
\( f(7)=7^{2}-7=49 - 7 = 42 \)

Answer:

\( 8 \)

2nd Problem