Question

the table shows a function. is the function linear or nonlinear?

x	y
\\(3\frac{2}{5}\\)	\\(4\frac{2}{5}\\)
9	10

options: linear, nonlinear

Explanation:

Step1: Recall linear function definition

A linear function has a constant rate of change (slope), i.e., $\frac{y_2 - y_1}{x_2 - x_1}=\frac{y_3 - y_2}{x_3 - x_2}$. First, convert mixed numbers to improper fractions.
$x_1=\frac{1}{5}$, $y_1 = 1\frac{1}{5}=\frac{6}{5}$;
$x_2=3\frac{2}{5}=\frac{17}{5}$, $y_2 = 4\frac{2}{5}=\frac{22}{5}$;
$x_3 = 9=\frac{45}{5}$, $y_3 = 10=\frac{50}{5}$.

Step2: Calculate first slope

Slope between $(x_1,y_1)$ and $(x_2,y_2)$:
$\frac{y_2 - y_1}{x_2 - x_1}=\frac{\frac{22}{5}-\frac{6}{5}}{\frac{17}{5}-\frac{1}{5}}=\frac{\frac{16}{5}}{\frac{16}{5}} = 1$.

Step3: Calculate second slope

Slope between $(x_2,y_2)$ and $(x_3,y_3)$:
$\frac{y_3 - y_2}{x_3 - x_2}=\frac{\frac{50}{5}-\frac{22}{5}}{\frac{45}{5}-\frac{17}{5}}=\frac{\frac{28}{5}}{\frac{28}{5}} = 1$.

Since the slopes are equal, the function has a constant rate of change.

Answer:

linear