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: Recall linear - function property

A linear function has the form $f(x)=mx + b$, and its rate of change (slope $m$) is constant. The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slopes between points

Let's take two - point pairs from the table. For the points $(-1,2)$ and $(0,2)$:
$m_1=\frac{2 - 2}{0-( - 1)}=\frac{0}{1}=0$.
For the points $(0,2)$ and $(1,2)$:
$m_2=\frac{2 - 2}{1 - 0}=\frac{0}{1}=0$.
For the points $(1,2)$ and $(2,2)$:
$m_3=\frac{2 - 2}{2 - 1}=\frac{0}{1}=0$.
Since the slope between any two points is $0$, the function is linear.

Step3: Find the linear - function form

Using the form $f(x)=mx + b$, with $m = 0$ and taking the point $(0,2)$ (when $x = 0$, $y = 2$), substituting into $y=mx + b$ gives $2=0\times0 + b$, so $b = 2$. Then $f(x)=0x+2=2$.

Answer:

A. $f(x)=2$