Question

find $g(3)$.
$g(x)$ is a line graph on a coordinate plane. also, there is a function $h(x) = 3x^2 - 2x + 2$ and a table for $f(x)$ with $x$ values 6, 4, -6, 2 and corresponding $f(x)$ values -9, -6, -4, -9. the main question is to find $g(3)$ using the graph of $g(x)$.

Explanation:

Step1: Identify the graph's nature

The graph of \( g(x) \) is a straight line, so it's a linear function. Let's find its equation. We can see two points: the y - intercept (when \( x = 0 \)) is \( - 5 \) (from the graph, the line crosses the y - axis at \( (0,-5) \)) and the x - intercept is \( (5,0) \).

Step2: Calculate the slope

The slope \( m \) of a line passing through two points \( (x_1,y_1)=(0, - 5) \) and \( (x_2,y_2)=(5,0) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-5)}{5 - 0}=\frac{5}{5}=1 \).

Step3: Write the equation of the line

Using the slope - intercept form \( y=mx + b \), where \( m = 1 \) and \( b=-5 \), the equation of \( g(x) \) is \( g(x)=x - 5 \).

Step4: Evaluate \( g(3) \)

Substitute \( x = 3 \) into \( g(x)=x - 5 \). So \( g(3)=3-5=-2 \).

Answer:

\(-2\)