A **relation** in mathematics is a set of pairings between **input** and **output** values. These pairs are often written as **coordinates** in the form \left(x, y\right), especially when relating to a graph in the coordinate plane.

A relation in which each input corresponds to exactly one output is known as a **function**.

Recall that a **function** maps each input of a **relation** to exactly one output. Functions are typically represented in **function notation**, so the relationship between inputs and outputs are clear.

The inputs and outputs of a function can also be described as the **domain** and **range**, which can be represented in different notations.

Domain: set of all possible inputs

Range: set of all possible outputs

Two common styles of notation for characteristics of functions include interval notation and set notation. **Interval notation** uses brackets and an **interval** to show the set of all numbers which lie between two values. **Set notation** specifies a set of elements that satisfy a set of conditions.

The following are graphs of functions with their domain and range written in both interval notation and set notation:

Consider the curve on the graph below.

a

State the domain of the function.

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b

State the range of the function.

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Consider the function shown in the graph:

a

State the domain.

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b

State the range.

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Consider the function shown in the graph:

a

Evaluate the function for f\left( -2 \right).

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b

Determine the value of x such that f\left( x \right) = -1.

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Let f\left( x \right) represent the height of a growing plant, f, in inches, where x represents the time since it was planted in days.

a

Interpret the meaning of f\left(10\right) = 8.

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b

Interpret the meaning of f\left(6\right).

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c

Interpret the meaning of f\left(x\right)=12.

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Idea summary

*Domain*: set of all possible inputs

*Range*: set of all possible outputs

*Interval Notation*: [-4, \infty)

*Set notation*: \{x| -4 \leq x < \infty\}

In addition to domain and range, characteristics for identifying and describing functions include:

Sections of functions can also display certain properties.

A connected region in the domain over which the output values become higher as the input values become higher is known as an **increasing interval**. Similarly, a connected region in the domain over which the output values (y-values) of a function become lower as the input values (x-values) become higher is known as a **decreasing interval**.

Increasing intervals: (-5, -2)\cup (0,4)

Decreasing interval: \left(-2,0\right)

Note that we did not use square brackets to include the endpoints of the intervals. This is because the function is not increasing or decreasing at the points of change between increasing and decreasing intervals. At these points, the function is considered to have a rate of change of zero.

If a function is neither increasing nor decreasing for part of its domain, we have a **constant interval**.

Consider the function shown in the graph:

a

Determine the coordinates of the absolute maximum or minimum.

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b

Determine the intervals where the function is increasing or decreasing.

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c

Write the domain and range of the function in interval notation.

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Consider the function shown in the graph:

a

Determine the equation of the asymptote.

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b

Identify the intercepts.

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c

Describe the end behavior of the function.

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Consider the shown function graph:

a

Determine the increasing and decreasing intervals.

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b

Determine any absolute and relative maxima and minima.

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For the given graph:

*Graph 1*

*Graph 2*

a

Compare the zeros of the two functions.

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b

Which graph has a domain of (-\infty, \infty)?

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c

Which function is only increasing?

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Idea summary

The characteristics of functions include:

domain and range

x- and y-intercepts

maximum or minimum value(s)

zeros

end behavior

increasing, decreasing, and constant intervals

asymptotes