Topics
1. Analyzing Functions
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Algebra 2

1.01 Characteristics of functions

Lesson
Practice

Functions, domain, and range

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.

Input

The independent variable of a function; usually the x-value

Output

The dependent variable of a function; usually the y-value

Function notation

A notation that describes a function. For a function f when x is the input, the symbol f\left(x\right) denotes the corresponding output.

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  • Consider f\left(2\right)

The 2 represents the x-value. To find the y-value we will start at 2 on the x-axis and find its corresponding y-value. Therefore, f\left(2\right)=4

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  • Consider the value(s) of x so that f\left(x\right)=4

The 4 represents the y-value. To find the x-value we will start at 4 on the y-axis and find it's corresponding x-value(s). Therefore, f\left(4\right)=-2, \text{ and } f\left(4\right)=2.

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

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
Domain: set of all possible inputs
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
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:

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Domain

\text{Interval Notation:}\,(-\infty,3)

\text{Set Notation:}\,\left\{x \middle\vert x < 3 \right\}

Range

\text{Interval Notation:}\,[-2,\infty)

\text{Set Notation:}\,\left\{y \middle\vert y \geq -2 \right\}

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Domain

\text{Interval Notation:}\,(-\infty,\infty)

\text{Set Notation:}\,\left\{x \middle\vert x \in \Reals \right\} \text{ or } \left\{x \middle\vert -\infty \lt x \lt \infty \right\}

Range

\text{Interval Notation:}\,[0,\infty)

\text{Set Notation:}\,\left\{y \middle\vert y \geq 0 \right\}

Examples

Example 1

Consider the curve on the graph below.

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a

State the domain of the function.

Worked Solution
Create a strategy

The domain is the set of x-coordinates of all the points on the curve. Notice the graph extends infinitely in both the positive and negative directions along the x-axis (to the left and right).

Apply the idea

Looking at the graph we see the function reaches all x-values, so our domain is all real numbers. In interval notation we can write this as:\text{Domain}=(-\infty, \infty)

Reflect and check

If we were to write our domain using set notation, we would have \{x | -\infty < x < \infty\}.

b

State the range of the function.

Worked Solution
Create a strategy

The range is the set of y-coordinates of all the points on the curve. This parabola has an absolute maximum at the vertex. There is no minimum y-value.

Apply the idea

Looking at the graph, the range of this function is all of the real numbers that are less than or equal to 2. This can be written in interval notation as:

\text{Range}=(-\infty , 2]

Reflect and check

If we were to write the range using set notation, we would have \{y |\, -\infty \lt y \leq 2\} .

Notice that 2 is included in the range because the graph of the function exists at that value. But we never include positive or negative infinity because it's not a number the graph can ever reach.

Example 2

Consider the function shown in the graph:

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a

State the domain.

Worked Solution
Create a strategy

Domain is the set of all possible x-values. We will look at the graph from left to right to identify all x-values for which the graph exists.

Apply the idea

The function reaches every x-value between -2 and 2, including -2 (notice the filled point) but not including 2 (notice the unfilled point).

We can show this in interval notation as:

Domain: [-2, 2)

Reflect and check

Let's now represent our domain using set notation.

Set Notation: \{x| -2 \leq x < 2\}

b

State the range.

Worked Solution
Create a strategy

Range is the set of all possible y-values. We will look at the graph from bottom to top to identify all y-values for which the graph exists.

Apply the idea

The function reaches every y-value between 0 (the lowest point on the graph, at the x-intercepts) and 4 (the highest point on the graph, at the vertex).

We can show that in interval notation as:

Range: [0, 4]

Reflect and check

The range in set notation is:

Set Notation: \{y \vert 0 \leq y \leq 4\}

Notice that 0 is included in the range because one of the points is filled. Even though the other point is unfilled, you only need one filled point on a graph to be able to include that value.

Example 3

Consider the function shown in the graph:

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a

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

Worked Solution
Create a strategy

We want to find the y-value when x=-2. We will start at -2 on the x-axis and move vertically until we hit the function. Then we will move horizontally to the y-axis and find the y-value there.

Apply the idea
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When x=-2, we can see on the graph the y-value is -4.

Reflect and check

If we knew the equation of the function, we could check by substituting -2 into the equation for x and confirming that the result when we evaluate is -4.

b

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

Worked Solution
Create a strategy

We want to find the x-value for when y=-1. We will start at -1 on the y-axis and move horizontally until we hit the function. Then we will move vertically to the x-axis and find the x-value there.

Apply the idea
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We can see from the graph that when the y-value is -1, the x-value is 7.

Reflect and check

This means that, f\left( 7 \right)=-1.

Example 4

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.

Plant Growth
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a

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

Worked Solution
Create a strategy

We can use the units of the given information and the graph to help with the interpretation.

Apply the idea

We're given that f\left( x \right) represents the height of a growing plant in inches, so to interpret f\left( 10 \right), we need to determine what an input of x = 10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted.

We also know that all of this is equal to 8. This is the output, or what our function f\left( x \right) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall.

Based on the graph, when x=10, y=8 so f\left(10\right)=8 is represented by the ordered pair (10,8) on the graph.

The plant has a height of 8 inches 10 days after being planted.

b

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

Worked Solution
Apply the idea

We know that x represents the time in days since the plant was planted and x=6. So this means that 6 days have passed since the plant was planted.

Since f\left( x \right) represents the height of a growing plant in inches, f\left( 6 \right) represents the height of the plant 6 days after being planted.

Reflect and check

Using the graph, we can find the actual height of the plant after 6 days.

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\text{Time (days)}
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c

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

Worked Solution
Apply the idea

We know that f\left( x \right) represents the height of a growing plant in inches, so if f\left( x \right)=12, then the height of the plant is 12 inches x days after being planted.

Reflect and check

By using the graph, we can find the number of days when the height of the plant is 12 inches.

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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\}

More characteristics of functions

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

x-intercept

A point (x,0) where a line or graph crosses the x-axis. The point represents the value of the domain where f\left(x\right)=0. A function can have any number of x-intercepts.

An x-intercept can also be called a zero.

The line y equals negative x plus 3 drawn in a coordinate plane. The point (3,0) is labeled x-intercept.
y-intercept

A point (x,0) where a line or graph crosses the y-axis. The point represents the value of the range where f\left(0\right)=y. A function can have at most one y-intercept.

The line y equals negative x plus 3 drawn in a coordinate plane. The point (0,3) is labeled y-intercept.
A curve plotted in a four quadrant coordinate plane. The curve resembles the letter m with the left peak higher than the right peak. The left peak is labeled absolute maximum, right peak labeled relative maximum, and the trough labeled relative minimum.

  • Absolute maximum: The point with the largest y-value across the domain

  • Absolute minimum: The point with the smallest y-value across the domain

  • Relative maximum: The point with the largest y-value in a region of the domain

  • Relative minimum: The point with the smallest y-value in a region of the domain

Asymptote

A line that a curve approaches as one or both of the variables in the equation of the curve approach infinity.

A decreasing exponential function approaching but never touching a dashed horizontal line labeled asymptote.
End behavior

Describes the trend of a function or graph at its left and right ends; specifically the y-value that each end obtains or approaches.

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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.

A graph made up of three connected line segments drawn in a coordinate plane. The first segment has endpoints at (negative 5, negative 2) and (negative 2, 4). The second segment has endpoints at (negative 2, 4) and (0, negative 4). The third segment has endpoints at (0, negative 4) and (4, 0). The first and third segment are colored green and labeled with increasing interval. The second segment is colored blue. On the x axis, the intervals negative 5 to negative 2 and 0 to 4 are colored green. The interval negative 2 to 0 is colored blue.
Increasing intervals: (-5, -2)\cup (0,4)
A graph made up of three connected line segments drawn in a coordinate plane. The first segment has endpoints at (negative 5, negative 2) and (negative 2, 4). The second segment has endpoints at (negative 2, 4) and (0, negative 4). The third segment has endpoints at (0, negative 4) and (4, 0). The first and third segment are colored green. The second segment is colored blue and labeled decreasing interval. On the x axis, the intervals negative 5 to negative 2 and 0 to 4 are colored green. The interval negative 2 to 0 is colored blue.
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.

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In the blue, left most, portion of the graph shown, the function is not increasing or decreasing. The y-value never increases or decreases over that interval, but instead remains the same. This is a constant interval.

Constant Interval: \left(-\infty, -5\right)

Examples

Example 5

Consider the function shown in the graph:

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a

Determine the coordinates of the absolute maximum or minimum.

Worked Solution
Create a strategy

For this function, the vertex is the absolute minimum. Note the values on the x-axis change by 1, and the values on the y-axis change by 2.

Apply the idea

The absolute minimum is the point (1,-4).

Reflect and check

Recall the axis of symmetry is a characteristic of quadratic functions. The axis of symmetry is the line that passes through the middle of the parabola and the x-value of the vertex. So, the axis of symmetry for this parabola is x=1.

b

Determine the intervals where the function is increasing or decreasing.

Worked Solution
Create a strategy

The increasing intervals are the domain values for which the values of f\left(x\right) increase as x increases. The decreasing intervals are the domain values where f\left(x\right) decreases as x increases.

Apply the idea

The function is increasing on the interval \left(1,\infty\right).

The function is decreasing on the interval \left(-\infty,1\right).

Reflect and check

Since the graph changes directions at a domain value of 1, and the graph does not have end points, we do not use square brackets for the intervals. Square brackets would only be used for defined endpoints of functions over a set domain.

c

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

Worked Solution
Create a strategy

Remember that the domain of the function is the set of all possible input values, which are the x-values that correspond to points on the graph.

Similarly, the range of the function is the set of all possible output values, which are the y-values that correspond to points on the graph.

Apply the idea

If we were to continue extending both ends of the function indefinitely, it would stretch upwards towards positive infinity on both sides. In addition, the ends would continue indefinitely toward the left and toward the right.

In part (a) we stated that the absolute minimum is the point (1,\,-4), this is also the vertex of the function.

So, the domain is "all real values" and the range is "all values greater than or equal to -4". In interval notation, this is:

  • Domain: \left(-\infty, \infty\right)

  • Range: \left[-4, \infty\right)

Reflect and check

In set notation, we would write the domain and range as follows:

  • Domain: \left\{x\vert x\in\Reals\right\} or \left\{x\vert -\infty < x < \infty \right\}
  • Range: \left\{y\vert y\geq -4\right\}

Example 6

Consider the function shown in the graph:

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a

Determine the equation of the asymptote.

Worked Solution
Create a strategy

Notice how the function approaches the x-axis without ever reaching it:

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Apply the idea

This means that the x-axis is an asymptote for the function. The equation of the horizontal aymptote is y = 0.

Reflect and check

Asymptotes are lines, so they should always be stated as an equation. The equation for horizontal asymptotes is always of the form y=c, and the equation for vertical asymptotes is always of the form x=c where c is any real number.

b

Identify the intercepts.

Worked Solution
Create a strategy

In part (a), we found there is a horizontal asymptote on the x-axis. This means there will be no x-intercepts, so we only need to identify the y-intercept.

Apply the idea

The y-intercept of the function is at \left(0,1\right).

Reflect and check

Recall from Algebra 1 that all the exponential functions of the form f\left(x\right)=b^{x} will have a y-intercept at \left(0,1\right). This is because f\left(0\right)=b^{0}=1. Later in this unit, we will begin transforming functions, so we will see exponential functions that do not have a y-intercept at \left(0,1\right).

c

Describe the end behavior of the function.

Worked Solution
Create a strategy
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The end behavior refers to the left and right "ends" of the graph. We want to know what happens to the output values as the input values get increasingly smaller (to the left) and what happens to the output values when the input values get increasingly larger (to the right).

Apply the idea

As the input values decrease indefinitely, the function output values continue to increase indefinitely. So as x \to -\infty, f\left(x\right)\to \infty.

As the input values increase indefinitely, the function output values approach the asymptote at y=0. So as x \to \infty, f\left(x\right)\to 0.

Example 7

Consider the shown function graph:

A curve plotted in a four quadrant coordinate plane. The curve has turning points at (negative 3.5, 6), (negative 1, negative 3),(1.5, 0.5), (4, negative 4), and (6,0). Speak to your teacher for more details.
a

Determine the increasing and decreasing intervals.

Worked Solution
Create a strategy

To determine the increasing and decreasing intervals, we need to first find the x-values of the turning points. These values will be the endpoints of the increasing and decreasing intervals.

The turning points occur at x=-3.5,-1,1.5,4,6.

Apply the idea

Tracing the graph from the smallest x-values toward the largest x-values, we can see that the graph switches between increasing and decreasing at each of the turning points.

Increasing intervals: \left(-\infty, -3.5\right)\cup \left(-1,1.5\right)\cup \left(4,6\right)

Decreasing intervals: \left(-3.5, -1\right)\cup \left(1.5,4\right)\cup \left(6,\infty\right)

Reflect and check

Notice that we are not intersted in the y-values when listing the increasing and decreasing intervals. If we used the y-values, some of them would be listed in multiple intervals which would make the notation confusing. By only using the domain values, the notation is clear and the x-values do not appear in multiple intervals.

b

Determine any absolute and relative maxima and minima.

Worked Solution
Create a strategy

We have already identified the x-values of the turning points in part (a). Now, we need to identify the corresponding y-values and classify each as an absolute maximum, an absolute minimum, a relative maximum, or a relative minimum.

Apply the idea

The point with the largest y-value is \left(-3.5, 6\right). This is the absolute maximum.

The relative maxima are \left(1.5, 0.5\right) and \left(6,0\right).

The relative minima are \left(-1, -3\right) and \left(4, -4\right).

There is no absolute minimum since the end behavior of the function tends toward negative infinity.

Example 8

For the given graph:

Graph 1

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Graph 2

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a

Compare the zeros of the two functions.

Worked Solution
Create a strategy

The zeros occur when the y-value is 0. In other words, the zeros are the x-values where the graph crosses the x-axis.

Apply the idea

Looking at Graph 1, we can see the y-value is 0 at the x-values 4 and -4, so the zeros are:

x=-4, 4

Looking at Graph 2, since it's a cube root function, it only has one zero at x=-4.

So, both functions share a zero of x=-4.

b

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

Worked Solution
Create a strategy

Remember that the domain of a function is the set of all possible input values, (x-values)

Apply the idea

Looking at Graph 1, we see the function reaches all x-values, so its domain is all real numbers.

Graph 2, being a cube root function, also reaches all x-values, since cube root functions are defined for all real numbers.

Both graphs have a domain of \left(-\infty, \infty\right).

c

Which function is only increasing?

Worked Solution
Create a strategy

The increasing intervals are parts of the domain where, when the y-values increase the x-values also increase.

Apply the idea

The function in Graph 1 is increasing from \left(3, \infty\right), constant from \left(-3, 3\right), and decreasing from \left(-\infty, -3\right).

Looking at the function in Graph 2, as we observe the y-values as we move from left to right, the y-values are always increasing. The function is increasing over the interval \left(-\infty, \infty \right).

Graph 2 is only increasing.

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

Outcomes

A2.F.2

The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

A2.F.2a

Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.

A2.F.2b

Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions.

A2.F.2c

Determine the intervals on which the graph of a function is increasing, decreasing, or constant.

A2.F.2d

Determine the location and value of absolute (global) maxima and absolute (global) minima of a function.

A2.F.2e

Determine the location and value of relative (local) maxima or relative (local) minima of a function.

A2.F.2f

For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable.

A2.F.2g

Describe the end behavior of a function.

A2.F.2h

Determine the equations of any vertical and horizontal asymptotes of a function using a graph or equation (rational, exponential, and logarithmic).

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