Question

1. which of the following equations represents a linear function? a. $y = 3x^2$ b. $y = 2 - \frac{6}{x}$ c. $y = \sqrt{x} + 6$ d. $y = \frac{1}{2}x + 3$

Explanation:

Step1: Recall linear function form

A linear function follows the form $y = mx + b$, where $m$ and $b$ are constants, and $x$ has an exponent of 1 (no higher powers, roots, or variables in denominators).

Step2: Analyze Option A

$y = 3x^2$ has $x$ raised to the power of 2, so it is quadratic, not linear.

Step3: Analyze Option B

$y = 2 - \frac{6}{x}$ can be rewritten as $y = 2 - 6x^{-1}$, where $x$ has a negative exponent, so it is not linear.

Step4: Analyze Option C

$y = \sqrt{x} + 6$ is equivalent to $y = x^{\frac{1}{2}} + 6$, where $x$ has a fractional exponent, so it is not linear.

Step5: Analyze Option D

$y = \frac{1}{2}x + 3$ matches the form $y = mx + b$ with $m=\frac{1}{2}$ and $b=3$, so it is linear.

Answer:

D. $y = \frac{1}{2}x + 3$