Question

Observe the given table and complete the following statement.

$x$	$y$
2	3
4	$k$
12	18

If the function represented in the table is nonlinear, then k will not be equal to ______.

• A. 3
• B. 6
• C. 5
• D. 7

Answer 1

The correct option is B 6

If the rate of change of a function remains constant between any three points, then it is a linear function. Else, it is a nonlinear function.

The rate of change of a function between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the expression $\frac{y_2-y_1}{x_2-x_1}$.

Let's find the rate of change.

$x$	$y$
2	3
4	$k$
12	18

$\underset{–}{\text{Point 1 and 2:}}$
Rate of change between the points $(2,3)$ and $(4,k)$
$=\frac{k-3}{4-2}$
$=\frac{k-3}{2}$

$\underset{–}{\text{Point 1 and 3:}}$
Rate of change between the points $(2,3)$ and $(12,18)$
$=\frac{18-3}{12-2}$
$=\frac{15}{10}$
$=\frac{3}{2}$

Given, the given function is nonlinear.
The rate of change between $(2,3)$ and $(4,k)\ne$ the rate of change between $(2,3)$ and $(12,18)$
$\Rightarrow \frac{k-3}{2}\ne \frac{3}{2}$
Multiplying both sides by 2,
$\Rightarrow \frac{k-3}{2}\times 2\ne \frac{3}{2}\times 2$
$\Rightarrow k-3\ne 3$
$\Rightarrow k\ne 6$

Hence, the value of k cannot be equal to 6 for the given function to be nonlinear.