Question

use the table to determine whether f(x) could represent a linear function. if it could, write f(x) in the form f(x)=mx + b. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. f(x)=□ (simplify your answer.) b. f(x) could not represent a linear function

Explanation:

Step1: Calculate the slope between points

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take the first two - points $(0,0)$ and $(2,8)$. Then $m_1=\frac{8 - 0}{2 - 0}=\frac{8}{2}=4$.
Take the second and third points $(2,8)$ and $(3,12)$. Then $m_2=\frac{12 - 8}{3 - 2}=\frac{4}{1}=4$.
Take the third and fourth points $(3,12)$ and $(5,20)$. Then $m_3=\frac{20 - 12}{5 - 3}=\frac{8}{2}=4$.
Since the slope between any two - points is the same ($m = 4$), it is a linear function.

Step2: Find the y - intercept

The equation of a line is $y=mx + b$. Using the point $(0,0)$ and $m = 4$, substitute $x = 0$, $y = 0$ and $m = 4$ into $y=mx + b$. We get $0=4\times0 + b$, so $b = 0$.
The function is $f(x)=4x$.

Answer:

A. $f(x)=4x$