Question

1. create a linear function using the given function evaluations.

$f(1) = 2$ and $f(3) = -4$

Explanation:

Step1: Recall linear function form

A linear function is of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. First, we need to find the slope \( m \) using the two points \((x_1,y_1)=(1,2)\) and \((x_2,y_2)=(3, - 4)\). The formula for slope is \( m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute the values: \( m=\frac{-4 - 2}{3 - 1}=\frac{-6}{2}=-3\).

Step2: Find the y - intercept \( b \)

Now that we have \( m=-3 \), we can use one of the points (let's use \((1,2)\)) and substitute into \( f(x)=mx + b \). So, \( 2=-3(1)+b \).
Solve for \( b \): \( 2=-3 + b\), then \( b=2 + 3=5\).

Step3: Write the linear function

Substitute \( m=-3 \) and \( b = 5 \) into the linear function form \( f(x)=mx + b \). We get \( f(x)=-3x + 5 \).

Answer:

\( f(x)=-3x + 5 \)