Question

1. determine whether each equation is a function or not a function:

	function	not a function
$y = 2$	$\square$	$\square$
$x^2 + y^2 = 9$	$\square$	$\square$
$y = 4x^4$	$\square$	$\square$
$4 - 2xy^2$	$\square$	$\square$
$x = 2$	$\square$	$\square$

Explanation:

To determine if an equation is a function, we use the vertical line test: a relation is a function if every vertical line intersects its graph at most once. A function must assign exactly one output (\(y\)-value) for each input (\(x\)-value).

1. \( y = 7x + 8 \)

This is a linear equation (slope-intercept form). For any \( x \), there is exactly one \( y \). Passes the vertical line test.
Classification: Function

2. \( y = 2 \)

This is a horizontal line. For any \( x \), \( y = 2 \) (only one \( y \)-value). Passes the vertical line test.
Classification: Function

3. \( x^2 + y^2 = 9 \)

This is the equation of a circle (radius 3, center at the origin). A vertical line (e.g., \( x = 0 \)) intersects the circle at two points (\( y = 3 \) and \( y = -3 \)). Fails the vertical line test.
Classification: Not a Function

4. \( y = 4x^4 \)

This is a polynomial equation. For any \( x \), there is exactly one \( y \) (since \( x^4 \) is non-negative, and multiplying by 4 and taking the 4th power gives a unique \( y \)). Passes the vertical line test.
Classification: Function

5. \( 4 = 2xy^2 \) (rewritten: \( y^2 = \frac{2}{x} \))

For a given \( x \) (e.g., \( x = 1 \)), \( y^2 = 2 \) implies \( y = \sqrt{2} \) or \( y = -\sqrt{2} \). Two \( y \)-values for one \( x \). Fails the vertical line test.
Classification: Not a Function

6. \( x = 2 \)

This is a vertical line. For \( x = 2 \), \( y \) can be any real number (infinite \( y \)-values for one \( x \)). Fails the vertical line test.
Classification: Not a Function

Final Classifications:

Equation	Function	Not a Function
\( y = 2 \)	✅	❌
\( x^2 + y^2 = 9 \)	❌	✅
\( y = 4x^4 \)	✅	❌
\( 4 = 2xy^2 \)	❌	✅
\( x = 2 \)	❌	✅

(Mark the appropriate boxes in the table with ✅ for the correct category.)

Answer:

To determine if an equation is a function, we use the vertical line test: a relation is a function if every vertical line intersects its graph at most once. A function must assign exactly one output (\(y\)-value) for each input (\(x\)-value).

1. \( y = 7x + 8 \)

This is a linear equation (slope-intercept form). For any \( x \), there is exactly one \( y \). Passes the vertical line test.
Classification: Function

2. \( y = 2 \)

This is a horizontal line. For any \( x \), \( y = 2 \) (only one \( y \)-value). Passes the vertical line test.
Classification: Function

3. \( x^2 + y^2 = 9 \)

This is the equation of a circle (radius 3, center at the origin). A vertical line (e.g., \( x = 0 \)) intersects the circle at two points (\( y = 3 \) and \( y = -3 \)). Fails the vertical line test.
Classification: Not a Function

4. \( y = 4x^4 \)

This is a polynomial equation. For any \( x \), there is exactly one \( y \) (since \( x^4 \) is non-negative, and multiplying by 4 and taking the 4th power gives a unique \( y \)). Passes the vertical line test.
Classification: Function

5. \( 4 = 2xy^2 \) (rewritten: \( y^2 = \frac{2}{x} \))

For a given \( x \) (e.g., \( x = 1 \)), \( y^2 = 2 \) implies \( y = \sqrt{2} \) or \( y = -\sqrt{2} \). Two \( y \)-values for one \( x \). Fails the vertical line test.
Classification: Not a Function

6. \( x = 2 \)

This is a vertical line. For \( x = 2 \), \( y \) can be any real number (infinite \( y \)-values for one \( x \)). Fails the vertical line test.
Classification: Not a Function

Final Classifications:

Equation	Function	Not a Function
\( y = 2 \)	✅	❌
\( x^2 + y^2 = 9 \)	❌	✅
\( y = 4x^4 \)	✅	❌
\( 4 = 2xy^2 \)	❌	✅
\( x = 2 \)	❌	✅

(Mark the appropriate boxes in the table with ✅ for the correct category.)