Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Your teacher will give you a set of cards that each contain a number. Decide whether each number is a possible input for the functions described here. Sort the cards into two groups: possible inputs and impossible inputs. Record your sorting decisions.

1. The area of a square, in square centimeters, is a function of its side length, $s$ , in centimeters. The equation $A (s) = s^{2}$ defines this function.

a. Possible inputs:

Compare your answers:

Possible inputs: $9$ , $\frac{3}{5}$ , $15$ , $0$ , $0.8$ , $4$ , $\frac{25}{4}$ , $0.001$ , $6.8$ , $72$

b. Impossible inputs:

Compare your answers:

Impossible inputs: –3, –18

2. A tennis camp charges $40 per student for a full-day camp. The camp only runs if at least 5 students sign up, and it limits the enrollment to 16 campers a day. The amount of revenue, in dollars, that the tennis camp collects is a function of the number of students who enroll. The equation $R (n) = 40 n$ defines this function.

a. Possible inputs:

Compare your answers:

Possible inputs: 9, 15

b. Impossible inputs:

Compare your answers:

Impossible inputs: $- 3$ , $- 18$ , $\frac{3}{5}$ , $0.8$ , $0$ , $\frac{25}{4}$ , $0.001$ , $4$ , $6.8$ , $72$

3. The relationship between temperature in Celsius and the temperature in Kelvin can be represented by a function $k$ . The equation $k (c) = c + 273.15$ defines this function, where $c$ is the temperature in Celsius and $k (c)$ is the temperature in Kelvin.

a. Possible inputs:

Compare your answer:

Possible inputs: All values

b. Impossible inputs:

Compare your answer:

Impossible inputs: No values

Write this definition in your math notebook: The domain of a function is the set of all of its possible input values.

Self Check

Jose created a function $W(h)$ for the weight, $W$ , of his family members given their height, $h$ , in inches.

Which of the following provides a group of possible inputs for the function?

$62$ , $-68$ , $70\frac34$
$47$ , $50.2$ , $60\frac12$
$5$ , $40$ , $62$
$36$ , $52$ , $-70$

Additional Resources

Finding Possible Input Values

1. Micah mows lawns every week where the function $M$ represents the number of lawns mowed as a function of $d$ , days, modeled by the equation $M (d) = 4 d$ . Select three of the following that are inputs for this function.

0, 2.5, and 4

2. Elena runs at a constant speed. Her distance, $D$ , in miles is a function of the time, $t$ , in minutes. Her running distance is represented by the function $D (t) = 9 t$ . Select three of the following that are inputs for this function.

40, $\frac{2}{3}$ , and 60.5

−7 is not possible because she cannot run for a negative amount of time

$\frac{2}{3}$ is possible. However, it is not always reasonable because it stands for $\frac{2}{3}$ of one minute. It is important to always look at the entire problem to see if there are any other restrictions, like a minimum amount of time running.

45 - 60.5 are both possible input values because she could run for 45 minutes or 60.5 minutes.

Using Inequalities to Determine Domain

The domain and range of a function describe the set of possible input values (domain) and output values (range) for that function. The domain represents the values that can be plugged into the function, Inequalities can be used to represent the domain. Remember to consider the nature of the function, its properties, and any specific restrictions when determining the domain using inequalities. Graphing the function can often provide visual clues and help confirm your analysis.

Use the following information to determine the domain of a function using inequalities:

Identify any restrictions or limitations on the input variable, usually denoted as $x$ .
Look for values of $x$ that could result in undefined or non-existent outputs.
Common restrictions include square roots of negative numbers, division by zero, or values excluded by the function's definition.

Use the following information to express the domain of a function using inequalities:

Use interval notation or set notation to represent the valid input values for the function.
Use inequalities to express any restrictions.

For example, if $x$ cannot be negative, you would write $(x \geq 0)$ or $(0, + \infty)$ in interval notation.

Special cases

Consider any specific exclusions or additional conditions mentioned in the function's definition.

For instance, if a function has a denominator and $x$ cannot equal certain values, denote those exclusions in the domain using the notation " $x$ ≠ value."

Example

Determine the domain of this function: $p (x) = 3, x \geq - 2 .$

The inequality $x \geq - 2$ indicates that the function is defined for all values of $x$ that are greater than or equal to $x \geq - 2$ .

Therefore, the domain of $p (x) = 3$ , $x \geq - 2$ is $x \geq - 2$ , or in interval notation, (−2, +∞).

Try it

Try It: Determine the Domain of a Function Using Inequalities

1. The distance a turtle has walked, in feet, after his race with the hare begins is represented by the function $f (x) = 0.5 x$ where $x$ is the time in seconds. For this scenario, what is the domain of the function $f (x) = 0.5 x$ using inequalities?

Compare your answer $x \geq 0$ is the domain written as an inequality. While the linear function $f (x) = 0.5 x$ has a domain of all real numbers, because the scenario describes the input as the time since the race began, it creates a set of limitations or restrictions on the domain.

The time the race starts begins at 0 seconds. Thus, $x$ must be greater than or equal to zero.
Time can be measured with decimal or fractional values, so $x$ can be any real number greater than or equal to zero.

The interval notation for this domain is $[0, + \infty)$ .

2. The profit a dance team makes from a fundraising raffle is represented by the function $f (x) = 3 x$ where $x$ represents the number of tickets sold. For this scenario, what is the domain of the function $f (x) = 3 x$ using inequalities?

Compare your answerThe domain written as an inequality is $x \geq 0$ , where $x$ is a whole number.

The dance team can sell 0, 1, 2 or more tickets, but they can't sell a negative number of tickets. Thus, $x$ must be greater than or equal to zero.
It does not make sense to sell half a raffle ticket, so $x$ must be a whole number.

The interval notation for this domain is $x$ is a whole number such that $[0, + \infty$ ).

3. What is the domain of the function $g (x) = \frac{1}{(x - 2)}$ , $x \neq 2$ using inequalities?

Compare your answer $x 2$ or $x > 2$ is the domain written with inequalities. To write the domain in interval notation, we use: $(- \infty, 2) \cup (2, + \infty)$ . The domain cannot include 2, so it is not part of the interval.

4.12.2 Domain: The Input of a Function

Activity

Additional Resources

Finding Possible Input Values

Using Inequalities to Determine Domain

Try It: Determine the Domain of a Function Using Inequalities