Question

The following functions represent the distance (y) of bus stand of 5 cities from their respective center points with respect to time taken (x).

$\text{Function for City 1:}y=2x+3$

$\text{Function for City 2:}y=\frac{2x}{3}+3$

$\text{Function for City 3:}y=3x^2-3$

$\text{Function for City 4: Between x = 0 and x = 5,}$
$\text{slope = 4. After x = 5,}$
$\text{slope keeps increasing.}$

$\text{Function for City 5: Between x = 0 and x = 6,}$
$\text{slope = 2.5. For all values of}$
$\text{x}\ge \text{7, slope = 2.5}$

How many functions of the given ones are linear?

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The correct option is B 3

For a function to be linear, the higest power (degree) of both the variables should be unity.

Let us consider function for City 1:

$y=2x+3$

Both the variables have power equal to 1. So, it is a linear function.

Let us consider function for City 2:

$y=\frac{2x}{3}+3$

Both the variables have power equal to 1. So, it is a linear function.

Let us consider function for City 3:

$y=3x^2-3$

Power of x is 2 here. So, it is not a linear function.



Let us consider function for City 4:

For a function to be linear, its slope should be constant. In this function, for x = 0 to x = 5, slope = 4. But, slope keeps increasing after x = 5. So, it is not a linear function.

Let us consider function for City 5:

For x = 0 to x = 6, slope = 2.5. For a function to be linear, the slope should be constant. For x ≥ 7, slope is constant. So, it is a linear function.

So, in total functions for City 1, 2, and 5 are linear ones.