Question

which of the following equations represent functions? assume x is the input and y is the output.
options:
$y = 7.5$
$y = 7x$
$y - x = 5$
$y = -x + 7$
$x = 75$

Explanation:

To determine which equations represent functions (where \( x \) is the input and \( y \) is the output), we use the vertical line test or the definition of a function: for every input \( x \), there is exactly one output \( y \).

Analyzing Each Equation:

1. \( \boldsymbol{y = 7.5} \):

For any \( x \), \( y \) is always \( 7.5 \). Each \( x \) maps to one \( y \) (constant function). Function.

1. \( \boldsymbol{y = 7x} \):

For any \( x \), \( y \) is uniquely determined by \( 7x \). Each \( x \) maps to one \( y \) (linear function). Function.

1. \( \boldsymbol{y - x = 5} \) (rewrite as \( y = x + 5 \)):

For any \( x \), \( y \) is uniquely determined by \( x + 5 \). Each \( x \) maps to one \( y \) (linear function). Function.

1. \( \boldsymbol{y = -x + 7} \):

For any \( x \), \( y \) is uniquely determined by \( -x + 7 \). Each \( x \) maps to one \( y \) (linear function). Function.

1. \( \boldsymbol{x = 75} \):

For \( x = 75 \), \( y \) is not defined (or could be any value). This fails the vertical line test (infinite \( y \)-values for one \( x \)). Not a function.

Answer:

The equations that represent functions are:

• \( y = 7.5 \)
• \( y = 7x \)
• \( y - x = 5 \) (or \( y = x + 5 \))
• \( y = -x + 7 \)