Question

which represents a non - linear function?
a. $y = \frac{1}{3}x + 5$

b. $y = - 7x + 8$

c. $y = \frac{1}{2}x^{2}+ 2$

d. $y = 3x - 8$

Explanation:

Step1: Recall linear function definition

A linear function has the form \( y = mx + b \), where \( m \) and \( b \) are constants, and the highest power of \( x \) is 1. Non - linear functions have a power of \( x \) other than 1 (e.g., quadratic, cubic, etc.) or other non - linear terms.

Step2: Analyze Option A

For \( y=\frac{1}{3}x + x^{2}\), the term \( x^{2} \) has a power of 2. So this function is quadratic (a type of non - linear function) since it has a term with \( x \) raised to the power of 2.

Step3: Analyze Option B

For \( y = - 7x+1\), it is in the form \( y=mx + b \) with \( m=-7 \) and \( b = 1 \). The highest power of \( x \) is 1, so it is a linear function.

Step4: Analyze Option C

Wait, there seems to be a typo. If we assume it's \( y=\frac{1}{2}x + 2\), it is in the form \( y=mx + b \) (linear). If it was supposed to be \( y=\frac{1}{2}x^{2}+2\), then it would be non - linear. But based on the original problem's likely intention and the other options, let's re - check. Wait, maybe the option C was miswritten. But among the given options as per the image (assuming the options are as presented), let's check Option D.

Step5: Analyze Option D

For \( y = 2x-8\), it is in the form \( y=mx + b \) with \( m = 2 \) and \( b=-8 \). The highest power of \( x \) is 1, so it is a linear function.

Answer:

A. \( y=\frac{1}{3}x + x^{2} \)