Question

cms 2025 - 2026 nc math 4 initial credit (approved for wmhs spr 26)
composition of functions and modeling
the function ( h(x)=\frac{4}{3}x - 1 ) represents the composite function ( h(x)=f(g(x)) ). if ( f(x)=2x - 1 ), what is ( g(x) )?
( g(x)=\frac{2}{3}x )
( g(x)=\frac{2}{3}x - 1 )
( g(x)=2\frac{1}{3}x - 1 )
( g(x)=2\frac{1}{3}x )

Explanation:

Step1: Set up composite function

We know $h(x)=f(g(x))$, $f(x)=2x-1$, so substitute $g(x)$ into $f$:
$h(x)=2g(x)-1$

Step2: Equate to given $h(x)$

Given $h(x)=\frac{4}{3}x - 1$, so:
$2g(x)-1=\frac{4}{3}x - 1$

Step3: Solve for $g(x)$

Add 1 to both sides:
$2g(x)=\frac{4}{3}x$
Divide both sides by 2:
$g(x)=\frac{4}{3x}\times\frac{1}{2}=\frac{2}{3}x$

Answer:

$g(x)=\frac{2}{3}x$