Question

consider the functions $f(x) = |x - 7|$ and $g(x) = 3x - 10$. what is the equation of $g(f(x))$? $g(f(x)) =$ select what is the value of $g(f(3))$? $g(f(3)) =$ select

Explanation:

Step1: Find \( g(f(x)) \)

To find the composition \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). Given \( f(x) = |x - 7| \) and \( g(x) = 3x - 10 \), we replace \( x \) in \( g(x) \) with \( f(x) \). So, \( g(f(x)) = 3f(x) - 10 \). Substituting \( f(x) = |x - 7| \) into this, we get \( g(f(x)) = 3|x - 7| - 10 \).

Step2: Find \( g(f(3)) \)

First, find \( f(3) \). Substitute \( x = 3 \) into \( f(x) \): \( f(3) = |3 - 7| = |-4| = 4 \). Then, substitute this result into \( g(x) \). So, \( g(f(3)) = g(4) \). Now, substitute \( x = 4 \) into \( g(x) \): \( g(4) = 3(4) - 10 = 12 - 10 = 2 \).

Answer:

For \( g(f(x)) \): \( 3|x - 7| - 10 \)
For \( g(f(3)) \): \( 2 \)