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4. Functions and Relations
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Algebra, Functions, and Data Analysis

4.04 Characteristics of functions

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Practice

Characteristics of functions

The important characteristics, or key features, of a function or relation include the previously seen domain, range, and the following additional features:

A parabola plotted in a four quadrant coordinate plane. The parabola opens upward, and passes through points (1, negative 4), (0, negative 3), (negative 1, 0), and (3, 0). The point (1, negative 4) is labeled minimum, (0, negative 3) labeled y-intercept, and (negative 1, 0) and (3, 0) labeled x-intercepts.

  • x-intercept(s) (also called zeros): the point(s) where a graph intersects the x-axis. A function can have multiple x-intercepts.

  • y-intercept: the point where a line or graph intersects the y-axis. A function can only have up to one y-intercept.

  • minimum: the lowest output value (absolute minimum) of a function across its entire domain.

  • maximum: the highest output value (absolute maximum) of a function across its entire domain.

The minimum is the single lowest point overall, and the maximum is the single highest point overall. These are often called the absolute minimum and absolute maximum.

These key features apply to a variety of functions, as we see from the following examples.

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  • Domain: \left\{x \middle\vert x \in \Reals \right\}

  • Range: \left\{y \middle\vert y \leq 4 \right\}

  • x-intercepts: \left(-1,0\right),\left(3,0\right)

  • Zeros: \left(-1,0\right),\left(3,0\right)

  • y-intercept: \left(0,3\right)

  • Maximum: \left(1,4\right)

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  • Domain: \left\{x \middle\vert -3 \leq x \leq 4 \right\}

  • Range: \left\{y \middle\vert -6 \leq y \leq 8 \right\}

  • x-intercept: \left(1,0\right)

  • Zeros: \left(1,0\right)

  • y-intercept: \left(0,2\right)

  • Maximum: \left(-3,8\right)

  • Minimum: \left(4,-6\right)

Examples

Example 1

Consider the function graphed.

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a

Does the function have a maximum or minimum value? If so, what is the value?

Worked Solution
Create a strategy

We need to find the lowest point on the graph and use the y-value to indicate how low it is.

Apply the idea
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This function has a minimum value of -4.

b

What is the range of the function?

Worked Solution
Create a strategy

In part (a), we identified that the function has a minimum value of -4. So we know that the function cannot take values smaller than -4.

Apply the idea

Looking at the function, we can see that it stretches up towards infinity on both sides of the minimum point. The function can take any value greater than or equal to -4.

The range of the function is:\text{Range: } \left\{y\, \vert\, y \geq -4\right\}

c

What are the x-intercept(s) of the function?

Worked Solution
Create a strategy

The x-intercept(s) of a function are the points where the function crosses the x-axis. In this case, by looking at the graph, we can see that there are two x-intercepts.

Apply the idea
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The x-intercepts of this function are the points \left(1, 0\right) and \left(5, 0\right).

Example 2

A penguin is tagged with a tracker to record its height above sea level when hunting. The height of the penguin is graphed against time.

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t\left(\text{mins}\right)
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Use the key features of the graph to describe the penguin's time spent hunting. Be as detailed as possible.

Worked Solution
Create a strategy

We can see that the graph has key features like intercepts, a minimum point, a domain, and a range.

To interpret the graph in context, we can use the axes of the graph to match key features to their real-world meaning.

Apply the idea

The y-intercept is \left(0,30\right). This means that the penguin is 30\text{ ft} above sea level at 0 minutes into its hunting time.

The x-intercepts are \left(5,0\right) and \left(20,0\right). This means that the penguin is at sea level at 5 minutes and 20 minutes into its hunting time.

The minimum of the graph is approximately \left(12.5,-41\right), so the penguin's lowest point is about 41\text{ ft} below sea level at about 12.5 minutes into its hunting time.

Combining this information gives a description of the penguin's hunting time. For example:

When the penguin needs to hunt, it leaves its nest, which is 30 \text{ ft} above sea level. The penguin makes its way down to the water and dives in 5 minutes after leaving the nest.

The penguin swims down to a depth of around 41\text{ ft} below sea level, reaching its deepest point around 12.5 minutes into its hunting time.

It then returns to the water's surface at 20 minutes. The penguin spends 15 minutes underwater in total.

The penguin then spends the last 5 minutes of its hunting time climbing back up to its nest, finishing a bit higher than where it started.

Reflect and check

There are many possible descriptions. We just need to ensure the description matches the key features of the graph.

For example, it is completely valid to say that the penguin dives into the water using a submarine as long as the deepest point is still 41\text{ ft} below sea level.

Example 3

The graph shows a hiker's elevation over a given period of time.

A first quadrant coordinate plane with x-axis labeled as Time and y-axis labeled as Elevation. A piecewise graph is plotted. Speak to your teacher for more information.
a

What could the zeros of the function represent in this situation?

Worked Solution
Create a strategy

The zeros of a function represent where f\left(x\right)=0.

Apply the idea

The y-axis represents elevation and the x-axis represents time.

The zeros of this function represent the times where the hiker reached ground level.

b

Describe the meaning of the maximum of the function in relation to the hiker's elevation over time.

Worked Solution
Create a strategy

The maximum is the highest point of the function.

Apply the idea

The function relates the elevation of the hiker over a period of time.

The maximum represents the time where the hiker reached their highest elevation on their journey.

c

Would the domain or range tell us how long the hiker traveled?

Worked Solution
Create a strategy

The domain represents the inputs, or x-values, of a function, while the range represents the outputs, or y-values.

Apply the idea

The x-axis is labeled as time.

The domain represents how long the hiker traveled.

Idea summary

The key features of a function and how to describe them are as follows:

  • We can write intercepts, zeros, minimums, and maximums as values or ordered pairs.

  • Domain and range: write in inequality, set notation, and interval notation.

More characteristics of functions

Additional key features for describing functions include asymptotes, end behavior, and intervals where the function increases, decreases, or remains constant.

Asymptote

A line that the graph of a function gets closer and closer to as the graph extends towards infinity (or sometimes towards a specific x-value for vertical asymptotes).

A decreasing exponential function approaching but never touching a dashed horizontal line labeled asymptote.
Vertical Asymptote

A vertical line x=c that the graph of a function approaches but never touches or crosses as the y-values approach positive or negative infinity. Vertical asymptotes occur at x-values where the function is undefined.

A graph showing two branches of a function approaching but not touching a dashed vertical line labeled vertical asymptote at x=c.
End behavior

Describes the trend of a function or graph at its left and right ends; specifically the y-value that each end gets closer and closer to (or heads towards, like infinity).

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Graphs can also show specific behaviors over certain sections or intervals.

An interval of x-values where the output (y-values) get higher as the input (x-values) get higher is known as an increasing interval. Similarly, an interval where the output values get lower as the input values get 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.

At these points, between increasing and decreasing intervals, 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, leftmost portion of the graph, 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)

Functions can have maximum and minimum values that are either absolute (the highest or lowest points on the entire graph) or relative (the highest or lowest points in a specific section of the graph, also called local maximums or minimums).

Absolute Maximum

The highest output value (y-value) of a function over its entire domain. A function may not have an absolute maximum if it increases indefinitely.

Absolute Minimum

The lowest output value (y-value) of a function over its entire domain. A function may not have an absolute minimum if it decreases indefinitely.

Relative Maximum

A point on the graph of a function that is higher than all nearby points. It represents the top of a "peak" in the graph. Also called a local maximum.

Relative Minimum

A point on the graph of a function that is lower than all nearby points. It represents the bottom of a "valley" in the graph. Also called a local minimum.

Examples

Example 4

Consider the function graphed:

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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), which is also the vertex of the function.

The domain is all real numbers, 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 5

Consider the function graphed:

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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 asymptote 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 that basic exponential functions like f(x)=b^x often have a y-intercept at (0,1).

This is because any base b raised to the power of 0 equals 1 (b^0=1). As we explore transformations later, we'll see how this intercept can change.

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).

We also want to know 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 6

Consider the function graphed:

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, \text{ and } 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 interested 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 7

For the given graphs:

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 of 4 and -4. 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, as the x-values increase, the y-values also increase.

Apply the idea

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

Looking at the function in Graph 2, as we observe 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.

Example 8

Consider the function graphed:

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a

Identify the equations of any vertical and horizontal asymptotes.

Worked Solution
Create a strategy

Look for dashed lines on the graph that the function approaches but does not cross. A vertical line indicates a vertical asymptote (x=c). A horizontal line indicates a horizontal asymptote (y=c).

Apply the idea

The graph shows a dashed vertical line at x=2. The function gets very close to this line on both sides but never touches it. So, the vertical asymptote is x=2.

The graph also shows a dashed horizontal line at y=1. The function approaches this line as x goes to positive and negative infinity. So, the horizontal asymptote is y=1.

b

State the domain and range of the function using interval notation.

Worked Solution
Create a strategy

The domain includes all possible x-values. Check if any x-values are excluded, often due to vertical asymptotes.

The range includes all possible y-values. Check if any y-values are excluded, often due to horizontal asymptotes or absolute max/min.

Apply the idea

The function exists for all x-values except where the vertical asymptote is, at x=2.

  • Domain: \left(-\infty, 2\right) \cup \left(2, \infty\right)

The function takes on all y-values except the value of the horizontal asymptote, y=1.

  • Range: \left(-\infty, 1\right) \cup \left(1, \infty\right)

c

Identify the intercepts.

Worked Solution
Create a strategy

The y-intercept is where the graph crosses the y-axis (x=0). The x-intercept (or zero) is where the graph crosses the x-axis (y=0).

Apply the idea

The y-intercept occurs where x=0. The graph passes through \left(0, 0.5\right).

The x-intercept occurs where y=0. The graph passes through \left(1, 0\right).

d

Describe the end behavior of the function.

Worked Solution
Create a strategy

Describe what happens to the y-values as x approaches negative infinity (far left) and as x approaches positive infinity (far right). Often, this relates to horizontal asymptotes.

Apply the idea

As x \to -\infty (moves left), the graph approaches the horizontal asymptote y=1. So, as x \to -\infty, f(x) \to 1.

As x \to \infty (moves right), the graph also approaches the horizontal asymptote y=1. So, as x \to \infty, f(x) \to 1.

e

Determine the intervals where the function is increasing or decreasing.

Worked Solution
Create a strategy

Read the graph from left to right. Identify the x-intervals where the graph is going downhill (decreasing) or uphill (increasing). Pay attention to breaks like vertical asymptotes.

Apply the idea

The graph is going downhill (decreasing) on both sides of the vertical asymptote.

The function is decreasing over its entire domain: \left(-\infty, 2\right) \cup \left(2, \infty\right). It is never increasing or constant.

Idea summary

Key characteristics help describe and understand functions graphed in the coordinate plane. We identify them visually:

  • Domain: Identify all x-values the graph covers from left to right. Note any breaks or excluded values (often at vertical asymptotes).

  • Range: Identify all y-values the graph covers from bottom to top. Note any excluded values (often at horizontal asymptotes or gaps).

  • y-intercept: Find the point where the graph crosses the y-axis (where x=0). A function has at most one.

  • x-intercept(s) / Zeros: Find the point(s) where the graph crosses the x-axis (where y=0). A function can have multiple.

  • Absolute Maximum/Minimum: Locate the overall highest/lowest point(s) on the entire graph.

  • Relative Maximum/Minimum: Locate the tops of "peaks" (relative max) and bottoms of "valleys" (relative min) in the graph.

  • Increasing/Decreasing/Constant Intervals: Read the graph from left to right. Identify the x-intervals where the graph goes uphill (increasing), downhill (decreasing), or stays level (constant). Use parentheses for interval endpoints unless the endpoint is included and the function stops there.

  • End Behavior: Describe the trend of the y-values as x approaches positive infinity (far right) and negative infinity (far left). Often involves approaching infinity or a horizontal asymptote.

  • Asymptotes: Look for lines (usually dashed) that the graph approaches but does not cross. Vertical asymptotes are x=c, horizontal are y=c. State their equations.

Outcomes

AFDA.AF.2a

Determine the domain and range of a function given a graphical representation, including those limited by contexts.

AFDA.AF.2b

Identify intervals on a graph for which a function is increasing, decreasing, or constant.

AFDA.AF.2c

Given a graph, identify the location and value of the absolute maximum and absolute minimum of a function over the domain of a function.

AFDA.AF.2d

Given a graph, determine the zeros and intercepts of a function.

AFDA.AF.2e

Describe and recognize the connection between points on the graph and the value of a function.

AFDA.AF.2f

Describe the end behavior of a function given its graph.

AFDA.AF.2g

Identify horizontal and/or vertical asymptotes from the graph of a function, if they exist.

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