Question

which statement is true regarding the functions on the graph?
$f(-3) = g(-4)$
$f(-4) = g(-3)$
$f(-3) = g(-3)$
$f(-4) = g(-4)$

Explanation:

Step1: Find equation of \( f(x) \)

The blue line \( f(x) \) has y - intercept \( 2 \) and slope \( m = \frac{8 - 2}{2 - 0}=3 \) (using points \((0,2)\) and \((2,8)\)). So \( f(x)=3x + 2 \).
For \( x=-3 \), \( f(-3)=3(-3)+2=-9 + 2=-7 \)? Wait, no, wait the graph: when \( x = - 3 \), let's check the intersection. Wait, maybe better to find \( g(x) \). The red line \( g(x) \): let's take two points. When \( x = 0 \), \( y=-8 \); when \( x=-4 \), let's see. Wait, actually, the two lines intersect at some point. Wait, maybe the correct way is to find the value of \( f(-3) \) and \( g(-4) \) from the graph. Wait, looking at the graph, the blue line \( f(x) \): when \( x=-3 \), what's \( y \)? Wait, the blue line passes through \((-1,0)\), \((0,2)\), so slope is \( 2 \) (wait, earlier mistake). Wait, from \((-1,0)\) to \((0,2)\), slope \( m=\frac{2 - 0}{0-(-1)} = 2 \). So \( f(x)=2x + 2 \). Then \( f(-3)=2(-3)+2=-6 + 2=-4 \).
Now \( g(x) \): the red line. Let's take two points: \((0,-8)\) and \((-4, - 4)\)? Wait, no, when \( x=-4 \), let's see the red line. Wait, the two lines intersect at \( x=-3 \)? Wait, no, the correct approach: let's find \( f(-3) \) and \( g(-4) \). Wait, maybe the first option: \( f(-3)=g(-4) \). Let's check \( f(-3) \): from the blue line, when \( x = - 3 \), let's see the y - value. The blue line has a slope of 2 (since from \( x=-1,y = 0 \) to \( x = 0,y = 2 \), slope is 2). So \( f(x)=2x + 2 \), so \( f(-3)=2*(-3)+2=-4 \). Now \( g(x) \): the red line. Let's find its equation. Let's take two points: when \( x = 0 \), \( y=-8 \); when \( x=-4 \), let's see, if we assume the red line has slope \( m=\frac{-4-(-8)}{-4 - 0}=\frac{4}{-4}=-1 \). So \( g(x)=-x - 8 \). Then \( g(-4)=-(-4)-8=4 - 8=-4 \). So \( f(-3)=-4 \) and \( g(-4)=-4 \), so \( f(-3)=g(-4) \).

Step2: Check other options

• For \( f(-4)=g(-3) \): \( f(-4)=2*(-4)+2=-8 + 2=-6 \); \( g(-3)=-(-3)-8=3 - 8=-5 \). Not equal.
• For \( f(-3)=g(-3) \): \( f(-3)=-4 \), \( g(-3)=-5 \). Not equal.
• For \( f(-4)=g(-4) \): \( f(-4)=-6 \), \( g(-4)=-4 \). Not equal.

Answer:

\( f(-3) = g(-4) \) (the first option with the blue circle, but as per the options, the correct statement is \( f(-3) = g(-4) \))