Question

question
find the value of f(7).
y = f(x)
(graph of a piecewise linear function with x-axis from -10 to 10 and y-axis from -10 to 10. one segment goes from (-8, 0) down to a vertex, then another segment goes up to cross the x-axis at (2, 0) and continues upwards. the question is to find f(7).)

Explanation:

Step1: Understand the graph

The graph of \( y = f(x) \) is given. To find \( f(7) \), we need to find the \( y \)-value when \( x = 7 \) on the graph.

Step2: Locate \( x = 7 \)

Find the point on the graph where \( x = 7 \). The right - hand line (the increasing part) of the piece - wise function passes through \( x = 7 \). We can determine the equation of the line or directly read the \( y \)-value from the graph.

Looking at the graph, the line that passes through \( x = 2 \) (where it crosses the \( x \)-axis) and has a slope. Let's check the slope. From \( x = 2 \) ( \( y = 0 \)) to \( x = 3 \), \( y = 1 \); \( x = 4 \), \( y = 2 \); and so on. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1-0}{3 - 2}=1 \). The equation of the line using point - slope form \( y - y_1=m(x - x_1) \) with \( (x_1,y_1)=(2,0) \) is \( y-0 = 1\times(x - 2) \), so \( y=x - 2 \).

When \( x = 7 \), substitute into the equation \( y=7 - 2=5 \). Alternatively, by looking at the grid, when \( x = 7 \), we can count the units up from the \( x \)-axis. Each grid square is 1 unit. Starting from \( x = 2,y = 0 \), for each increase of 1 in \( x \), \( y \) increases by 1. So when \( x=7 \), \( y=7 - 2 = 5 \).

Answer:

\( 5 \)