Task Sets: Graph root and piecewise functions - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

What is the domain of $f(x) = \sqrt{x}$ ?

A.

All real numbers

B.

$x \geq 0$ , written as $[0, \infty)$

C.

$x > 0$ , written as $(0, \infty)$

D.

$x \neq 0$

2.

What is the domain of $g(x) = \sqrt[3]{x}$ ?

A.

$x \geq 0$

B.

$x \neq 0$

C.

All real numbers

D.

$x > 0$

3.

A function is defined as:
$f(x) = \begin{cases} 2x & \text{if } x \leq 3 \\ 10 & \text{if } x > 3 \end{cases}$
What is $f(3)$ ?

A.

6

B.

10

C.

Both 6 and 10 — it depends on which piece you use

D.

Undefined at $x = 3$

B

Fluency Practice

1.

Which of the following is a strategic point for graphing $f(x) = \sqrt{x}$ ? (Choose the option where all three points lie on the graph.)

A.

$(0, 0)$ , $(1, 1)$ , $(4, 2)$

B.

$(0, 0)$ , $(2, 4)$ , $(3, 9)$

C.

$(-4, -2)$ , $(0, 0)$ , $(4, 2)$

D.

$(0, 0)$ , $(1, 2)$ , $(4, 4)$

2.

Which statement correctly compares the parent functions $y = \sqrt{x}$ and $y = \sqrt[3]{x}$ ?

A.

Both have domain $[0, \infty)$ and range $[0, \infty)$ .

B.

$y = \sqrt{x}$ has domain $[0, \infty)$ ; $y = \sqrt[3]{x}$ has domain $(-\infty, \infty)$ .

C.

Both graphs pass through the origin but only $y = \sqrt{x}$ is always increasing.

D.

$y = \sqrt[3]{x}$ has domain $[0, \infty)$ and $y = \sqrt{x}$ has domain $(-\infty, \infty)$ .

3.

Find three strategic points for $f(x) = 2\sqrt{x - 3} - 1$ and state the domain. (Hint: use inputs that make $x - 3$ a perfect square.)

4.

Find three strategic points for $h(x) = \sqrt[3]{x + 1} - 2$ and state the domain.

5.

Consider the piecewise function:
$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 4 & \text{if } x \geq 1 \end{cases}$
(a) Evaluate $f(0)$ , $f(1)$ , and $f(3)$ .
(b) Is the function continuous at $x = 1$ ? Explain.
(c) State the domain and range.

C

Varied Practice

D

Word Problems

1.

A parking garage charges a flat fee based on time parked. The cost function is:
$C(t) = \begin{cases} 3 & \text{if } 0 < t \leq 1 \\ 5 & \text{if } 1 < t \leq 2 \\ 7 & \text{if } 2 < t \leq 3 \end{cases}$
where $t$ is time in hours and $C$ is cost in dollars.

(a) What is the cost for parking 1.5 hours? 2 hours?
(b) What is the cost for parking exactly 2.0 hours? 2.1 hours?
(c) How should the endpoints be drawn on the graph of this function?

2.

A cell phone plan charges different rates:
$C(x) = \begin{cases} 30 & \text{if } 0 \leq x \leq 500 \\ 30 + 0.05(x - 500) & \text{if } x > 500 \end{cases}$
where $x$ is data used in megabytes and $C$ is cost in dollars.

1.

The cost for using 500 MB is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ dollars. The cost for using 600 MB is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ dollars.

cost at 500 MB:

cost at 600 MB:

2.

Is this function continuous at $x = 500$ ? Verify algebraically.

3.

The function $f(x) = |x - 3| + 1$ models the distance from a target of 3, shifted up by 1.

(a) Rewrite $f(x) = |x - 3| + 1$ as a piecewise-defined function.
(b) Find the vertex and state whether the graph is a V-shape or a smooth curve.
(c) Find $f(0)$ and $f(5)$ .

4.

A ball drops from a tower. The time to fall $d$ feet is given by $t(d) = \sqrt{d/16}$ seconds.

Find the time to fall 4 ft, 16 ft, and 64 ft. State the domain of $t$ in context and explain why negative $d$ values are excluded.

E

Error Analysis

1.

A student is asked to describe the graph of $f(x) = |x|$ . The student writes: "The graph is a U-shape, like a parabola, with a smooth bottom at the origin. Both sides curve upward."

What is wrong with the student's description?

A.

The graph of $|x|$ is a V-shape with a sharp corner at the origin — not a smooth U-shape. The two sides are straight lines, not curves.

B.

The graph of $|x|$ is correct as a smooth U-shape; the student is right.

C.

The graph should extend to the left of the origin, not both directions.

D.

The graph opens downward, not upward.

2.

A student graphs:
$f(x) = \begin{cases} x + 1 & \text{if } x \leq 2 \\ 3x - 5 & \text{if } x > 2 \end{cases}$
At $x = 2$ , the student draws two solid dots: one at $(2, 3)$ from the first piece and one at $(2, 1)$ from the second piece.

What error did the student make?

A.

The second piece uses $x > 2$ (strict), so its graph should start with an open circle at $(2, 1)$ , not a solid dot. Only the first piece includes $x = 2$ , so $(2, 3)$ is solid and $(2, 1)$ is open.

B.

The first piece should have an open circle at $(2, 3)$ because $x \leq 2$ is not strict.

C.

Both solid dots are correct — the function is defined at $x = 2$ by both pieces.

D.

The student should connect the two dots with a vertical line segment.

F

Challenge / Extension

1.

Write a three-piece piecewise function whose graph forms a "Z" shape: a horizontal segment at $y = 4$ for $x \leq 0$ , a diagonal segment with slope $-1$ for $0 \leq x \leq 4$ , and a horizontal segment at $y = 0$ for $x \geq 4$ . State the function and specify the endpoint convention at $x = 0$ and $x = 4$ .

2.