Question

which of the following relations represent y as a function of x? select all that apply.

(image of a mapping diagram with x - values a, b, c, d, e and y - values 2, 3, 4, 5, 6)

(equation: $3y^2 = 9x^2 + 5$)

(table with x: -2, -1, 0, 0, 2, 3 and y: 7, 7, 8, 7, 7, 7)

(image of an ellipse on a coordinate plane)

Explanation:

Step1: Check mapping diagram

A function requires each $x$-value maps to one $y$-value. In the left mapping, every element in the $x$-set ($a,b,c,d,e$) connects to exactly one $y$-value. This satisfies the function definition.

Step2: Check equation $3y^2=9x^2+5$

Solve for $y$:
$$y^2 = 3x^2 + \frac{5}{3}$$
$$y = \pm\sqrt{3x^2 + \frac{5}{3}}$$
Each $x$-value gives two $y$-values, so this is not a function.

Step3: Check table of values

For $x=0$, there are two different $y$-values ($8$ and $7$). A single $x$-value cannot map to multiple $y$-values, so this is not a function.

Step4: Check graph (horizontal ellipse)

Use the vertical line test: a vertical line drawn through the ellipse intersects the graph at two points. This means one $x$-value corresponds to two $y$-values, so this is not a function.

Answer:

The only relation that represents $Y$ as a function of $X$ is the leftmost mapping diagram (the oval set with elements $a,b,c,d,e$ mapping to $2,3,4,5,6$).