Question

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

If the function represented in the table is nonlinear, then $k$ can not be equal to:

• A. $6$
• B. $3$
• C. $5$
• D. $7$

Answer 1

The correct option is A $6$

If the rate of change of a function is the same between any two points, then it is linear; otherwise, it is nonlinear.

A function's rate of change between two points $(x_1,y_1)$ and $(x_2,y_2)$ is: $\frac{y_2-y_1}{x_2-x_1}$

Let's find the rate of change:

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

Row 1 and 2:

Rate of chage between $(2,3)$ and $(4,k)$
$=\frac{k-3}{4-2}$
$=\frac{k-3}{2}$

Row 1 and 3:

Rate of chage between $(2,3)$ and $(12,18)$
$=\frac{18-3}{12-2}$
$=\frac{15}{10}$
$=\frac{5\times 3}{5\times 2}$
$=\frac{3}{2}$

Given, the given function is nonlinear.
$\therefore$ Rate of change between $(2,3)$ and $(4,k)\ne$ Rate of chage between $(2,3)$ and $(12,18)$
$\Rightarrow \frac{k-3}{2}\ne \frac{3}{2}$

Multiply both sides by 2
$\Rightarrow \frac{k-3}{2}\times 2\ne \frac{3}{2}\times 2$
$\Rightarrow k-3\ne 3$

Add 3 to both sides
$\Rightarrow k-3+3\ne 3+3$
$\Rightarrow k\ne 6$

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