Question

decide which of these relations represents ( y ) as a function of ( x ).
( square -8y = x + 9 )
( square x = y^2 - 6 )
( square -7x - 6y = 5 )
( square y = x^2 )
( square ) none of these are functions

Explanation:

To determine if a relation represents \( y \) as a function of \( x \), we use the vertical line test: for each \( x \)-value, there should be at most one \( y \)-value. We can also solve for \( y \) and check if each \( x \) gives a unique \( y \).

Step 1: Analyze \( -8y = x + 9 \)

Solve for \( y \):
\[
-8y = x + 9 \implies y = -\frac{x + 9}{8} = -\frac{1}{8}x - \frac{9}{8}
\]
This is a linear equation (slope-intercept form \( y = mx + b \)). For any \( x \), there is exactly one \( y \). So this is a function.

Step 2: Analyze \( x = y^2 - 6 \)

Solve for \( y \):
\[
x = y^2 - 6 \implies y^2 = x + 6 \implies y = \pm\sqrt{x + 6}
\]
For a given \( x > -6 \), there are two \( y \)-values (positive and negative square roots). For example, if \( x = 3 \), \( y = \pm 3 \). So this is not a function.

Step 3: Analyze \( -7x - 6y = 5 \)

Solve for \( y \):
\[
-7x - 6y = 5 \implies -6y = 7x + 5 \implies y = -\frac{7}{6}x - \frac{5}{6}
\]
This is a linear equation. For any \( x \), there is exactly one \( y \). So this is a function.

Step 4: Analyze \( y = x^2 \)

This is a quadratic equation. For any \( x \), squaring it gives exactly one \( y \)-value (e.g., \( x = 2 \) gives \( y = 4 \), \( x = -2 \) gives \( y = 4 \); note that multiple \( x \)-values can give the same \( y \), but the vertical line test only requires one \( y \) per \( x \), not vice versa). So this is a function.

Answer:

The relations that represent \( y \) as a function of \( x \) are:

• \( -8y = x + 9 \)
• \( -7x - 6y = 5 \)
• \( y = x^2 \)

(To mark the checkboxes: check \( -8y = x + 9 \), \( -7x - 6y = 5 \), and \( y = x^2 \).)