Question

for each part, determine if y is a function of x.
(a) a scientist is running an experiment. she adjusts the temperature of a solution so that it freezes and then measures some variable. let x be the temperature, and y be the time (in seconds) after the start of the experiment, at which the solution is a dry solid.
(circ) y is a function of x. (circ) y is not a function of x.
(b) brian walks from his house to the post office, and then returns home. y is the time (in minutes) after he left, and x is his distance from home.
(circ) y is a function of x. (circ) y is not a function of x
(c) a yo - yo moves up and down 5 times. y is the yo - yos distance above the ground x seconds after it starts moving.
(circ) y is a function of x. (circ) y is not a function of x.

Explanation:

Part (a)

To determine if \( y \) is a function of \( x \), we use the definition of a function: for each input \( x \) (temperature), there should be exactly one output \( y \) (time when the solution is at that temperature). In an experiment where the solution's temperature is adjusted to freeze and melt, at a given temperature \( x \) (during freezing or melting), there should be only one time \( y \) when the solution is at that temperature (assuming the process is well - defined and the temperature - time relationship for a given temperature during the freeze - melt cycle has a unique time associated with it). So, for each \( x \), there is one \( y \).

For \( y \) to be a function of \( x \), each \( x \) (distance from home) must correspond to exactly one \( y \) (time after leaving). When Brian walks to the post office and back, for a given distance \( x \) (other than 0 and the maximum distance to the post office), there are two times: one when he is going to the post office and one when he is coming back. For example, if the distance to the post office is \( d \), at \( x=\frac{d}{2} \), there is a time when he is on his way to the post office and a different time when he is on his way back. So, a single \( x \) can correspond to two different \( y \) values, so \( y \) is not a function of \( x \).

Using the function definition, for each \( x \) (seconds after starting to move), the yo - yo's distance above the ground \( y \) is uniquely determined. The yo - yo's motion (up and down) is a well - defined motion, and at a specific time \( x \), the yo - yo is at exactly one distance above the ground. So, for each input \( x \), there is exactly one output \( y \).

Answer:

\( y \) is a function of \( x \).

Part (b)