$\DeclareMathOperator{\asin}{asin}$ $\DeclareMathOperator{\acos}{acos}$ $\DeclareMathOperator{\atan}{atan}$

The animation below shows values of x being related to values of y by a relationship named f. Play the video, then answer the questions following the video using meanings for terms in the question. You can replay the video and pause it, or scroll the play bar to examine parts of the video. (Move your pointer away from the video to remove the scroll bar.)

1. What, approximately, is the domain of f?

The value of x appears to vary through all values from -7 to 7. Each of these values is related to a value of y. The domain of f is therefore all values of x such that $-7\le x \le 7$.

2. What, approximately, is the range of f?

The value of y appears to range from -9 to 9 as the value of x varies from -7 to 7. It does not matter that y takes on the same value for different values of x.

3. Is the relationship named f a function from x to y? Explain.

Yes, f is a function from x to y. Each value of x in the domain of f is related to exactly one value of y.

4. Is f continuous over its domain? Explain.

It appears from the video that f is continuous over its domain. The value of y varies smoothly, with no gaps, as the value of x varies smoothly in its domain.

5. Is f one-to-one over its domain? Explain.

f is not 1-1 over its domain. You can see this by focusing on the value of x as it varies from -2 to 2. f is 1-1 up to $x=-2$; then the value of y varies over values it has already taken on in relation to smaller values of x. In particular, $f(-4)=0$ and $f(0)=0$, so f is not 1-1.

6. Over what interval(s) in its domain, if any, is f increasing? Explain.

It appears $f(u)\lt f(v)$ when $u\lt v$ for u and v in the domain of f and both are less than -2. It also appears $f(u)\lt f(v)$ when $u\lt v$ for u and v in the domain of f and both are greater than 2.

Another way to say this is the value of y increases as the value of x increases from -7 to -2 and from 2 to 7.

7. Over what interval(s) in its domain, if any, is f decreasing? Explain.

It appears $f(u)\gt f(v)$ when $u\lt v$ for u and v in the domain of f and both are between -2 and 2.

Another way to say this is the value of y decreases as the value of x increases from -2 to 2.