Question

the graph of $y = f(x)$ is shown below. find the value of $f(1)$. answer attempt 1 out of 2 + additional solution - no solution

Explanation:

Step1: Recall the meaning of \( f(1) \)

To find \( f(1) \), we need to determine the \( y \)-value of the function \( y = f(x) \) when \( x = 1 \). This means we look at the point on the graph of \( y = f(x) \) where the \( x \)-coordinate is 1 and find the corresponding \( y \)-coordinate.

Step2: Analyze the graph

Looking at the graph of the line \( y = f(x) \), we can see that it is a straight line. To find the value at \( x = 1 \), we can either use the equation of the line or directly observe the \( y \)-value at \( x = 1 \) from the graph.

First, let's find the equation of the line. The line passes through the points \( (0, -3) \) (the \( y \)-intercept) and \( (3, 0) \) (the \( x \)-intercept). The slope \( m \) of the line is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the points \( (0, -3) \) and \( (3, 0) \):
\( m=\frac{0 - (-3)}{3 - 0}=\frac{3}{3} = 1 \)

So the equation of the line in slope-intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the \( y \)-intercept) is \( y = 1x-3 \) or \( y=x - 3 \).

Now, substitute \( x = 1 \) into the equation:
\( y=f(1)=1 - 3=-2 \)

Alternatively, by looking at the graph, when \( x = 1 \), we can see that the \( y \)-value is -2 (since the line passes through \( (1, -2) \)).

Answer:

\( -2 \)