Question

click on all the functions that are not linear. a. $y = 6(2 - x)$ b. $y = 8x$ c. $y = 9 - \frac{1}{2}x^2$ d. $y = 4 - 7x$ e. $y = \frac{3}{5x} - 3$ f. $y = 5(x^2 + 2)$ g. $y = \frac{4}{5}x + 10$

Explanation:

Step1: Recall linear function definition

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

Step2: Analyze Option A

Simplify the expression:
$y=6(2-x)=12-6x$, which fits $y=mx+b$. Linear.

Step3: Analyze Option B

$y=8x$ fits $y=mx+b$ (with $b=0$). Linear.

Step4: Analyze Option C

$y=9-\frac{1}{2}x^2$ has $x^2$ (exponent 2). Not linear.

Step5: Analyze Option D

$y=4-7x$ fits $y=mx+b$. Linear.

Step6: Analyze Option E

$y=\frac{3}{5x}-3=\frac{3}{5}x^{-1}-3$, $x$ is in the denominator (exponent -1). Not linear.

Step7: Analyze Option F

Simplify the expression:
$y=5(x^2+2)=5x^2+10$, has $x^2$ (exponent 2). Not linear.

Step8: Analyze Option G

$y=\frac{4}{5}x+10$ fits $y=mx+b$. Linear.

Answer:

C. $y=9-\frac{1}{2}x^2$, E. $y=\frac{3}{5x}-3$, F. $y=5(x^2+2)$