Question

homework 13: section 3.1
score: 57.5/260 answered: 6/26
question 7
decide which of these relations represents y as a function of x.
□ ( y = x^2 )
□ ( y = 10x^2 - 2x )
□ ( x = y )
□ ( -4x + y^2 = 18 )
□ none of these are functions

Explanation:

To determine if a relation represents \( y \) as a function of \( x \), we use the vertical line test: a relation is a function if every vertical line intersects the graph at most once.

Step 1: Analyze \( y = x^2 \)

For any input \( x \), squaring it gives exactly one output \( y \). For example, if \( x = 2 \), \( y = 4 \); if \( x = -2 \), \( y = 4 \). Even though different \( x \)-values can give the same \( y \)-value, each \( x \) has only one \( y \), so it passes the vertical line test. Thus, \( y = x^2 \) is a function.

Step 2: Analyze \( y = 10x^2 - 2x \)

This is a quadratic equation in \( x \). For any real number \( x \), substituting into the equation gives exactly one \( y \)-value. For instance, if \( x = 1 \), \( y = 10(1)^2 - 2(1) = 8 \); if \( x = 0 \), \( y = 0 \). Each \( x \) maps to one \( y \), so it passes the vertical line test and is a function.

Step 3: Analyze \( x = y \)

We can rewrite this as \( y = x \). For every \( x \), there is exactly one \( y \) (equal to \( x \)). So it passes the vertical line test and is a function.

Step 4: Analyze \( -4x + y^2 = 18 \)

Rewrite it as \( y^2 = 4x + 18 \), so \( y = \pm\sqrt{4x + 18} \). For a given \( x \) (where \( 4x + 18 \geq 0 \), i.e., \( x \geq -\frac{9}{2} \)), there are two possible \( y \)-values (positive and negative square roots). For example, if \( x = 0 \), \( y^2 = 18 \), so \( y = \sqrt{18} \) or \( y = -\sqrt{18} \). This fails the vertical line test, so it is not a function.

Answer:

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

• \( y = x^2 \)
• \( y = 10x^2 - 2x \)
• \( x = y \)

(So the correct options are the checkboxes next to \( y = x^2 \), \( y = 10x^2 - 2x \), and \( x = y \).)