Question

There are $4$ functions given below.
Function 1: $\frac{y}{3}=7x-3$

Function 2:

$x$	$3$	$5$	$8$	$12$
$y$	$5$	$9$	$15$	$23$

Function 3:

Function 4:
Kylie's cab company charges a one-time pickup fee of $\$5$ for every ride, as well as a $\$2.90$ charge for each mile traveled.
___________________________ represent(s) a linear function.

• A. All the functions
• B. Function 1
• C. Function 1 and Function 2
• D. Function 1, Function 2, and Function 3

Answer 1

The correct option is A All the functions

A function can be considered a linear function if it can be represented as $y=mx+b$, where $m$ represents the slope and $b$ represents the $y-$intercept.
Also, a function should have a constant rate of change to qualify as a linear function.
Let us consider all the functions one by one.

Function 1:

$\frac{y}{3}=7x-3$
$\Rightarrow \frac{y}{3}\times 3=3\times (7x-3)$
$\Rightarrow y=21x-9$
$\Rightarrow y=21x+(-9)$
Here, the function is rewritten in the slope-intercept form $y=mx+b$.
So the function is a linear function.

Function 2:
Here, the rate of change $=\frac{\text{change in y values}}{\text{change in x values}}$
$=\frac{4}{2}\text{or}\frac{6}{3}\text{or}\frac{8}{4}$
$=2$
Here, the function has a constant rate of change.
So the function is a linear function.

Function 3:

The given graph is a straight line.
We know, the graphical representation of any linear function is a straight line.

So the function is a linear function.

Function 4:
Kylie's cab company charges a one-time pickup fee of $\$5$ for every ride, as well as a $\$2.90$ charge for each mile traveled.
Given that a one-time pickup fee is $\$5$,
Charge for each mile traveled is $\$2.90$.

Let us consider, $x=$ number of miles covered
$y=$ total charges of cab
So the function for total charges by Kylie's cab company $=$
$y=5+2.90+2.90+\dots \dots \dots +uptoxdays$
$\Rightarrow y=5+2.90x$
$\Rightarrow y=2.90x+5$
Here, the above expression can be expressed as $y=mx+b$.

So the given function is a linear function.

So option D is the correct answer.