Q: A hospital wants to model the distance between two ambulances leaving at right angles, one at 35 mph and the other at 40 mph. After 2 hours, what is the distance between them?

$$\sqrt{(35 \times 2)^2$ + (40 \times 2)^2$} = \sqrt{70^2$ + 80^2$} = \sqrt{4900 + 6400} = \sqrt{11300} \approx 106.3$$ miles

Question 1

Given the function $$f(x)$$ defined as follows:

$$\begin{cases} x+1, & x  3 \end{cases}$$

What is the value of $$f(-3)$$?

Accepted Answer

$$-2$$

Question 2

Which of the following best describes a function $$f$$ that is constant on an interval $$I$$?

Accepted Answer

For all $$a$$ and $$b in$$ $$I$$, $$f(a) = f(b)$$

Question 3

Two nurses, Kiara and Matias, drive away from a hospital at right angles to each other. Kiara’s speed is 35 mph and Matias’s is 40 mph. What is the distance between them after $$t$$ hours?

Accepted Answer

$$d(t) = \sqrt{(35t)^2$ + (40t)^2$}$$

Question 4

For the function $$f(x)$$ defined as $$f(x) = x^2$ for x$$ \u003e 3$$, what is the domain of this piece of the function?

Accepted Answer

$$x \u003e 3$$

Question 5

Which of the following intervals describes where the function $$f(x) is $$increasing if for all $$a \u003c b in$$ $$I$$, $$f(a) \u003c f(b)$$?

Accepted Answer

$$I is an $$interval where $$f is $$increasing

Question 6

Suppose $$f(x)$$ has a relative maximum at $$x = c. $$Which statement must be true?

Accepted Answer

There exists an open interval containing $$c$$ such that $$f(c) \u003e f(x)$$ for all $$x in $$the interval where $$x 
eq c$$

Question 7

Evaluate the greatest integer function $$f(x) = \lfloor x \rfloor at$$ $$x = 0.99.$$

Accepted Answer

$$0$$

Question 8

A function $$g(x) is $$defined piecewise as $$g(x) = \frac{1}{3}x + 3$$ for $$x \u003c 3$$ and $$g(x) = -x$$ for $$x \geq 3. $$What is $$g(4)$$?

Accepted Answer

$$-4$$

Question 9

Which of the following is the correct domain for the function $$d(t) = \sqrt{(35t)^2$ + (40t)^2$}$$ modeling the distance between two nurses driving away from a hospital at right angles?

Accepted Answer

$$[0, \infty)$$

Question 10

Given the piecewise function $$f(x) = \begin{cases} x-2, & x 
eq -2 \ 3, & x = -2 \end{cases}$$, what is the value of $$f(-2)$$?

Accepted Answer

$$3$$

Question 11

If a function $$f is $$decreasing on an interval $$I$$, which of the following must be true for all $$a \u003c b in$$ $$I$$?

Accepted Answer

$$f(a) \u003e f(b)$$

Question 12

Suppose a function $$h(x) is $$defined as $$h(x) = x+1$$ for $$x  3. $$What is $$h(10)$$?

Accepted Answer

$$100$$

Question 13

Which of the following is NOT a property of the greatest integer function $$f(x) = \lfloor x \rfloor$$?

Accepted Answer

For $$x = 0.25$$, $$\lfloor x \rfloor = 1$$

Question 14

A function $$f(x)$$ has a relative minimum at $$x = c. $$Which of the following statements is true?

Accepted Answer

There exists an open interval containing $$c$$ such that $$f(c) \u003c f(x)$$ for all $$x in $$the interval where $$x 
eq c$$

Question 15

Consider the function $$f(x) = \lfloor x \rfloor. $$What is $$f(-0.25)$$?

Accepted Answer

$$-1$$

Question 16

Which of the following best describes a piecewise function?

Accepted Answer

A function defined by different formulas on different parts of its domain

Question 17

If $$f(x) is $$defined as $$f(x) = x+1$$ for $$x  3$$, what is $$f(3)$$?

Accepted Answer

$$5$$

Question 18

A hospital wants to model the distance between two ambulances leaving at right angles, one at 35 mph and the other at 40 mph. After 2 hours, what is the distance between them?

Accepted Answer

$$\sqrt{(35 	imes 2)^2$ + (40 	imes 2)^2$} = \sqrt{70^2$ + 80^2$} = \sqrt{4900 + 6400} = \sqrt{11300} \approx 106.3$$ miles

Precalculus: Increasing, Decreasing, and Piecewise Functions

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