Question

the graph of a function h is shown below. find h(3).

Explanation:

Step1: Understand the problem

We need to find the value of the function \( h \) at \( x = 3 \), i.e., \( h(3) \), from the given graph of the function.

Step2: Analyze the graph

The graph is a straight line (linear function). To find \( h(3) \), we look for the point on the graph where \( x = 3 \) and then determine the corresponding \( y \)-value (since for a function \( y = h(x) \), the \( y \)-coordinate at \( x = 3 \) is \( h(3) \)).

First, let's find the equation of the line to confirm or directly read from the graph. Let's find the slope \( m \) and the \( y \)-intercept \( b \).

From the graph, when \( x = 0 \), \( y = 2 \), so the \( y \)-intercept \( b = 2 \).

To find the slope, we can use two points. Let's take the \( y \)-intercept \( (0, 2) \) and another point, say \( (-2, 0) \) (where the line crosses the \( x \)-axis).

The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{0 - (-2)}=\frac{2}{2}=1 \).

So the equation of the line is \( y = mx + b = x + 2 \), so \( h(x)=x + 2 \).

Step3: Calculate \( h(3) \)

Substitute \( x = 3 \) into the function \( h(x)=x + 2 \):

\( h(3)=3 + 2 = 5 \).

Alternatively, from the graph, when we move to \( x = 3 \) on the \( x \)-axis, we can see that the corresponding \( y \)-value is 5 (by moving up from \( x = 3 \) until we hit the line and then left to the \( y \)-axis, we reach \( y = 5 \)).

Answer:

\( h(3)=\boxed{5} \)