Q: A tax system is modeled by the piecewise function $$ T(x) = \begin{cases} 0.1x & x \u003c 10000 \\ 1000 + 0.2(x-10000) & x \geq 10000 \end{cases} $$ What is the tax on an income of $$15,000$$?

$$T(15000) = 1000 + 0.2 \times (15000 - 10000) = 1000 + 1000 = 2000$$

Q: Suppose a medication is administered at a rate modeled by $$ R(t) = \begin{cases} 0 & t \u003c 1 \\ 10 & 1 \leq t \u003c 3 \\ 5 & t \geq 3 \end{cases} $$ What is the total amount of medication administered from $$t = 0 to$$ $$t = 4$$ hours?

$$10 \times 2 + 5 \times 1 = 25$$

Q: Which of the following is the correct way to write a piecewise function for $$f(x)$$ that equals $$x$$ for $$x \u003c 0$$ and $$x^2$$ for $$x \geq 0$$?

$$f(x) = \begin{cases} x & x \u003c 0 \\ x^2$ & x \geq 0 \end{cases}$$

Question 1

Which of the following best describes a piecewise function?

Accepted Answer

A function made up of multiple equations, each applying to a different interval of the domain.

Question 2

Given the piecewise function
$$
f(x) = \begin{cases} -x & 	ext{if } x \u003c -1 \ x^2$ - 4 & 	ext{if } x \geq -1 \end{cases}
$$
What is $$f(-3)$$?

Accepted Answer

$$f(-3) = -(-3) = 3$$

Question 3

For the same function
$$
f(x) = \begin{cases} -x & 	ext{if } x \u003c -1 \ x^2$ - 4 & 	ext{if } x \geq -1 \end{cases}
$$
What is $$f(-1)$$?

Accepted Answer

$$f(-1) = (-1)^2$ - 4 = -3$$

Question 4

Evaluate $$f(2)$$ for the function
$$
f(x) = \begin{cases} -x & 	ext{if } x \u003c -1 \ x^2$ - 4 & 	ext{if } x \geq -1 \end{cases}
$$

Accepted Answer

$$f(2) = 2^2$ - 4 = 0$$

Question 5

A piecewise function is defined as
$$
f(x) = \begin{cases} 4 & x \u003c 3 \ x - 3 & 3 \leq x \u003c 8 \ x + 5 & x \geq 8 \end{cases}
$$
What is $$f(2)$$?

Accepted Answer

$$f(2) = 4$$

Question 6

Using the same function as above, what is $$f(5)$$?

Accepted Answer

$$f(5) = 5 - 3 = 2$$

Question 7

For the function
$$
f(x) = \begin{cases} 4 & x \u003c 3 \ x - 3 & 3 \leq x \u003c 8 \ x + 5 & x \geq 8 \end{cases}
$$
What is $$f(8)$$?

Accepted Answer

$$f(8) = 8 + 5 = 13$$

Question 8

Consider the piecewise function
$$
f(x) = \begin{cases} -4 & x \u003c -2 \ x + 1 & -2 \leq x \u003c 1 \ x^2$ & x \geq 1 \end{cases}
$$
What is $$f(-2)$$?

Accepted Answer

$$f(-2) = -2 + 1 = -1$$

Question 9

For the same function as above, what is $$f(1)$$?

Accepted Answer

$$f(1) = 1^2$ = 1$$

Question 10

A hospital uses a piecewise function to calculate the cost $$C(x) (in $$dollars) for $$x$$ hours of parking:
$$
C(x) = \begin{cases} 0 & x \u003c 1 \ 5 & 1 \leq x \u003c 4 \ 5 + 2(x-4) & x \geq 4 \end{cases}
$$
How much does a patient pay for 6 hours of parking?

Accepted Answer

$$C(6) = 5 + 2(6-4) = 9$$

Question 11

Which of the following statements about piecewise functions is TRUE?

Accepted Answer

Piecewise functions can have jumps or discontinuities at the boundaries between pieces.

Question 12

A tax system is modeled by the piecewise function
$$
T(x) = \begin{cases} 0.1x & x \u003c 10000 \ 1000 + 0.2(x-10000) & x \geq 10000 \end{cases}
$$
What is the tax on an income of $$15,000$$?

Accepted Answer

$$T(15000) = 1000 + 0.2 	imes (15000 - 10000) = 1000 + 1000 = 2000$$

Question 13

Given the function
$$
f(x) = \begin{cases} 2x + 4 & x \u003c 0 \ x^2$ & x \geq 0 \end{cases}
$$
Find all $$x$$ such that $$f(x) = 4.$$

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 14

A piecewise function is defined as
$$
f(x) = \begin{cases} x^2$ & x \u003c 2 \ 3x - 4 & x \geq 2 \end{cases}
$$
For which value(s) of $$x is$$ $$f(x)$$ not continuous?

Accepted Answer

$$x = 2$$

Question 15

Suppose a medication is administered at a rate modeled by
$$
R(t) = \begin{cases} 0 & t \u003c 1 \ 10 & 1 \leq t \u003c 3 \ 5 & t \geq 3 \end{cases}
$$
What is the total amount of medication administered from $$t = 0 to$$ $$t = 4$$ hours?

Accepted Answer

$$10 	imes 2 + 5 	imes 1 = 25$$

Question 16

Which of the following is the correct way to write a piecewise function for $$f(x)$$ that equals $$x$$ for $$x \u003c 0$$ and $$x^2$$ for $$x \geq 0$$?

Accepted Answer

$$f(x) = \begin{cases} x & x \u003c 0 \ x^2$ & x \geq 0 \end{cases}$$

Question 17

A company charges for shipping as follows: $$\5$ for packages under 2$ kg, $$\$$10$$ for $$2 kg to$$ $$5 kg$$, and $$\10 + 2(x-5$$)$$ for packages over 5$ kg, where x$ is the weight in kg. What is the cost to ship a 7$ kg package?

Accepted Answer

$$10 + 2(7-5) = 14$$

Question 18

If a piecewise function has a 'jump' at $$x = a$$, what does this mean?

Accepted Answer

The left and right limits at $$x = a$$ are not equal.

Study Guide: Piecewise Functions (Calculus Prerequisites)

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