Question

Which of the following shows a nonlinear function?

Answer 1

We know the standard form of a linear function in slope$-$intercept form is $y=mx+b$, where $m$ is the slope or rate of change and $b$ is the $y-$intercept.
That means any function that can be represented in the above form, will be considered a linear function.
Let us consider all the functions one by one.

$\to$ Option A: $y-5=2(x+1)$
$\Rightarrow y-5=2x+2$ (use distributive property)
$\Rightarrow y=2x+2+5$
$\Rightarrow y=2x+7$

The given equation is rewritten in the slope-intercept form.
By comparing the above equation with $y=mx+b,m=2\text{and}b=7$.

$\to$ Option B: $y=7^2\times x$
$\Rightarrow y=49\times x$
$\Rightarrow y=49x$
$\Rightarrow y=49x+0$
By comparing the above equation with $y=mx+b,m=49\text{and}b=0$.

$\to$ Option C: $y=\frac{x}{4}$
$\Rightarrow y=\frac{1}{4}x$
$\Rightarrow y=\frac{1}{4}x+0$
By comparing the above equation with $y=mx+b,m=\frac{1}{4}\text{and}b=0$.

$\to$ Option D: $y=\frac{-2}{x}$
$\Rightarrow y=-2\times x^{-1}$
$\Rightarrow y=-2x^{-1}$
Here, the above equation cannot be compared with $y=mx+b$, as the power of $x$ is different.

So option D is the correct answer.