Question

select the correct answer from each drop-down menu.
write the functions represented by the graphs.
f(x) =
g(x) =
h(x) =

Explanation:

Step1: Analyze \( g(x) \)

\( g(x) \) passes through the origin \((0,0)\) and \((2,1)\). The slope \( m = \frac{1 - 0}{2 - 0} = \frac{1}{2} \). Since it's an absolute - value function (V - shaped), and for \( x\geq0 \), \( y=\frac{1}{2}x \); for \( x < 0 \), \( y =-\frac{1}{2}x \). So \( g(x)=\frac{1}{2}|x| \).

Step2: Analyze \( f(x) \)

\( f(x) \) passes through \((0,0)\) and has a steeper slope than \( g(x) \). Let's take a point, say when \( x = 2 \), \( y = 2 \) (from the graph). The slope \( m=\frac{2-0}{2 - 0}=1 \). So for \( x\geq0 \), \( y = x \); for \( x < 0 \), \( y=-x \). Thus \( f(x)=|x| \).

Step3: Analyze \( h(x) \)

\( h(x) \) has a y - intercept at \( (0,3) \) and the same slope as \( f(x) \) (since the steepness is the same as \( f(x) \)). The general form of an absolute - value function is \( y = a|x|+b \). Here \( a = 1 \) (same slope as \( f(x) \)) and \( b = 3 \) (y - intercept). So \( h(x)=|x| + 3 \).

Answer:

\( f(x)=\boldsymbol{|x|} \), \( g(x)=\boldsymbol{\frac{1}{2}|x|} \), \( h(x)=\boldsymbol{|x| + 3} \)