Task Sets: Relate domain to graph and context - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

Which of the following correctly expresses "all real numbers greater than or equal to 3" in interval notation?

A.

$( 3, \infty )$

B.

$[ 3, \infty )$

C.

$( -\infty, 3 ]$

D.

$( -\infty, 3 )$

2.

Which expression is undefined for some real number value of $x$ ?

A.

$f(x) = 3x - 7$

B.

$g(x) = x^2 + 4$

C.

$h(x) = \dfrac{1}{x + 2}$

D.

$k(x) = |x| + 1$

3.

A graph of a function has an open circle at $( 3, 5 )$ . What does this mean?

A.

The function is undefined for all $x$ near 3.

B.

The point $( 3, 5 )$ is excluded — $x = 3$ is not in the domain.

C.

The function has a vertical asymptote at $x = 3$ .

D.

The output at $x = 3$ is zero.

B

Fluency Practice

C

Varied Practice

D

Word Problems

1.

A candle is 12 inches tall when lit. It burns at a steady rate of 0.5 inches per hour. The function $L(t) = 12 - 0.5t$ gives the candle's length in inches after $t$ hours.

State the natural domain and the contextual domain of $L$ . Explain the difference.

2.

A factory assembles engines. The function $h(n) = 12n + 5$ gives the number of person-hours needed to assemble $n$ engines.

1.

What is the natural domain of $h$ ? Write your answer in interval notation.

2.

What is the contextual domain of $h$ ? Use set-roster or set-builder notation and explain your reasoning.

3.

A bus sells tickets at $2.50 each. The revenue function is $R(p) = 2.50p$ where $p$ is the number of passengers. The bus holds at most 40 passengers.

Give the contextual domain of $R$ and state whether it is discrete or continuous. Justify your answer.

4.

A ball is launched upward. Its height in feet is given by $h(t) = -16t^2 + 32t + 6$ , where $t$ is time in seconds. The ball lands when $h(t) = 0$ . Using the quadratic formula, the positive landing time is approximately $t \approx 2.17$ seconds.

Give the contextual domain of $h$ in interval notation and explain why both endpoints are chosen.

E

Error Analysis

1.

A student graphs $R(n) = 9n$ , the revenue from selling $n$ tickets at $9 each, where $n$ must be a whole number from 0 to 5. The student draws a solid line from $(0, 0)$ to $(5, 45)$ .

What error did the student make?

A.

The student connected the points with a line, but the domain is discrete — only whole-number values of $n$ are valid. The graph should show 6 isolated dots at $n = 0, 1, 2, 3, 4, 5$ .

B.

The student used the wrong slope. The slope should be 1, not 9.

C.

The student should have drawn the line extending beyond $n = 5$ , since the natural domain is all reals.

D.

The student plotted the wrong starting point — the line should start at $(1, 9)$ , not $(0, 0)$ .

2.

A student is asked for the domain of $h(n) = 12n + 5$ , where $n$ is the number of engines assembled in a factory. The student writes: "Domain: $(-\infty, \infty)$ — all real numbers, because the formula works for any value of $n$ ."

What did the student get right, and what did the student overlook?

A.

The student correctly found the natural domain but overlooked the contextual domain. In context, $n$ must be a positive integer.

B.

The student is wrong — the natural domain of $h$ is not all real numbers.

C.

The student is completely correct — the contextual domain does not matter.

D.

The student should have found $n \geq 0$ as the natural domain instead.

F

Challenge / Extension

1.

A piecewise function is defined as:
$f(x) = \begin{cases} \sqrt{x + 4} & \text{if } x < 1 \\ \dfrac{1}{x - 3} & \text{if } x \geq 1 \end{cases}$
Find the domain of $f$ and write it in interval notation. Show all steps.

2.