Topics
4. Functions & Linear Relationships
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United States of AmericaFL
Grade 912

4.02 Domain and range

Lesson
Practice
Lesson

Concept summary

Two defining parts of any function are its domain and range.

A curved function graphed on a coordinate plane with an unfilled endpoint at the left most part of the function and a filled endpoint at the right most part of the function. A segment across the top of the function marks the domain with an unfilled enpoint to the left and a filled endpoint to the right

The set of all possible input values (x-values) for a function or relation is called the domain.

In the example shown, the domain is the interval of values -3 < x \leq 1. Notice that -3 is not included in the domain, which is indicated by the unfilled point.

A curved function graphed on a coordinate plane with an unfilled endpoint at the left most part of the function and a filled endpoint at the right most part of the function. A vertical segment to the right of the function marks the range with filled endpoints at the top and bottom

The set of all possible output values (y-values) for a function or relation is called the range.

In the example shown, the range is the interval of values -4 \leq y \leq 0. Notice that both endpoints are included in the range, since the function reaches a height of y = 0 at the origin.

A domain which is made up of disconnected values is said to be a discrete domain.

A four quadrant coordinate plane with seven points plotted in different quadrants. The points are located at (negative 3, 0), ( negative 54, 3),(negative 2.5, negative 1), (0, negative 3), (1.5, negtaive 2.5), (3, 0) and (3.5, 2).

A function with a discrete domain. It is only defined for distinct x-values.

A domain made up of a single connected interval of values is said to be a continuous domain.

A four quadrant coordinate plane plotted with a parabola, a continuous curve that opens upward. The parabola has its vertex at (0, negative 3), and passes through (negative 3, 0) and (3, 0)

A function with a continuous domain. It is defined for every x-value in an interval.

It is possible for the domain of a function to be neither discrete nor continuous.

The domain and range of a function are commonly expressed using inequality notation or set-builder notation.

Note that if two functions have different domains, then they must be different functions, even if they take the same values on the shared parts of their domains.

Worked examples

Example 1

Consider the function shown in the graph.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
a

State whether the function has a discrete or continuous domain.

Solution

The function is defined at every value of x across an interval, so it has a continuous domain.

b

Determine the domain of the function using set-builder notation.

Solution

We can see that the function is defined for every x-value between -6 and 8, including -6 but not including 8.

So the domain of the function can be written as \text{Domain: } \left\{x\, \vert\, -6 \leq x < 8\right\}

c

Determine the range of the function using set-builder notation.

Solution

We can see that the function reaches every y-value between -6 and 4, including 4 but not including -6.

So the range of the function can be written as \text{Range: } \left\{y\, \vert\, -6 < y \leq 4\right\}

Outcomes

MA.912.F.1.2

Given a function represented in function notation, evaluate the function for an input in its domain. For a real-world context, interpret the output.

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