Domain and Range of Functions

Multiple Choice

The student council sent its members on four field trips during the school year. The number of buses needed to transport the members on each trip is a function of the number of members who went on each trip. This function consists only the ordered pairs (52, 3), (72, 4), (86, 5), and (105, 6). What is the domain for this situation?

• {52, 105}

• {3, 4, 5, 6}

• {52, 72, 86, 105}

• {3, 4, 5, 6, 52, 72, 86, 105}

Multiple Choice

For the function {(0,1), (1,-3), (2,-4), (-4,1)}, write the domain and range.

• D: {1, -3, -4,}
  R: {0, 1, 2, -4}
• D: {-4, 0, 1, 2}
  R:{-4, -3, 1}
• D: {0, 1, 2, 3, 4}
  R:{1, -3, -4}
• D: {0, 1, 2, -4}
  R: {1, -3, -4, 1}

​What is a Relation?

​A "relation" is a pairing of input values with output values.

​The set of input values is called the domain and the set of output values is called the range.

​Find the domain and range of this relation

​Practice!

​Functions

For every function, there is only one y-value per x-value (or, one output per input). So,

if you can plug in an

input and get two

different answers,

then the relation is

not a function.​

​Functions

We can identify functions one of two ways: checking if any x-value has two or more

​y-values, or using the

​vertical line test on

​relation graphs.

​

​In the vertical line test,

​if you draw a vertical

​line through the graph

​and hit two points on the graph, then it is not a function.

​Is the above relation a function?

​Practice!

Multiple Choice

Identify the domain of the relation.

• -9, -5, 1, 7

• 1, 4, 5, 8

• -9, -5, 1, 1, 7

• 4, 8

​Relations in Two Variables

​An equation in two variables can relate two values like in the equation: y = 2x + 3. The input variable, x, is the independent variable and the output value, y, is the dependent variable (since its value depends on the value of the input).

​Graphing an Equation

​To graph an equation in two variables, we select values for x, plug them in to the equation, and then solve for y. This gives us a coordinate pair (x, y) that we can plot on a Cartesian plane.

​

​The relation to the right is a function and is linear.

​

​NOTE: When working with functions, we sometimes use function notation which is written f(x) = mx + b, and f(x) stands in place of y.

NOTE:​

Functions which contain variables with exponents are not linear.

Examples: y = 4x³ - 8, and y = 2x² + 4x - 16 are not linear since one x-value in each has an exponent attached to it.​

Domain and Range

• Domain (left to right)

• Range (bottom to top)

Domain (from left to right)

Multiple Choice

Which algebraic notation represents the DOMAIN?

• $$\left[-3,1\right]$$  

• $$-3<x\le1$$  

• $$\left(-3,1\right)$$  

• $$-3\le x<1$$  

Range (from bottom to top)

Multiple Choice

Which interval notation represents the Range?

• $$\left[-2,4\right]$$  

• $$-2\le y\le4$$  

• $$\left(-2,4\right)$$  

• $$-2\le y<4$$  

Arrows mean infinity

Arrows mean infinity

Multiple Choice

Which is the correct DOMAIN interval notation?

• (-2,4]

• [-2,4)

• $$-2<x\le4$$

• $$-2\le x<4$$

Multiple Choice

Which is the correct RANGE algebraic notation?

• $$-2\le y\le3$$  
  

• $$\left[-2,3\right]$$  
  

• $$\left[-5,3\right]$$

• $$-5\le y\le3$$

Discontinuous Domain

Multiple Choice

What is the DOMAIN of the graph?

• $$y\ge-3$$

• $$\left[-\infty,\infty\right]$$

• $$\left(-\infty,\infty\right)$$

Multiple Choice

What is the RANGE of the graph?

• $$y\ge-3$$

• $$\left[-\infty,\infty\right]$$

• $$\left(-\infty,\infty\right)$$

• $$\left[\infty,3\right]$$  

Multiple Choice

The amount of money that Li spends on video games depends on how many games he purchases. If Li has at most $90 to spend on games for his Xbox 360 and the games cost $22.50 each, what is the domain of this function?

• $$0\le x\le90$$

• $$0\le x\le4$$

• $$\left\{0,1,2,3,4\right\}$$

Multiple Choice

Thomas opened a savings account with $250. He plans to add $200 each week over the 12 week summer break. He will close the account at the end of summer. Which is an appropriate DOMAIN?

• $$x\ge0$$

• $$0\le x\le2650$$

• $$0\le x\le12$$

• $$250\le x\le2650$$

Multiple Choice

Thomas opened a savings account with $250. He plans to add $200 each week over the 12 week summer break. He will close the account at the end of summer. Which is an appropriate RANGE?

• $$y\ge0$$

• $$0\le y\le2650$$

• $$0\le y\le12$$

• $$250\le y\le2650$$

Multiple Choice

Which is the correct domain of the graph provided?

• $$\left(-\infty,-1\right)U\left(2,\infty\right)$$

• $$\left(-\infty,-1\right]U\left[2,\infty\right)$$

• $$\left(-\infty,\infty\right)$$