Functions and Graphs

Properties of a Function

Even and Odd Functions

Functions can be identified as even, odd, or neither. An even function is a line of symmetry about the yy-axis. An odd function has rotational symmetry about the origin.

A function can be even, odd, or neither. The graph of a function that is even or odd has a type of symmetry. If a graph has symmetry with respect to a line, then the part of the graph on one side of the line is a mirror image of the other half of the line. If a graph has symmetry with respect to a point, then the image of the graph is the same after it has been turned by 180º.

Some examples of even functions are f(x)=xnf(x)=x^n, where nn is any even number. Some examples of odd functions are f(x)=xnf(x)=x^n, where nn is any odd number.

Knowing whether a function is even or odd can be useful in graphing because one half of the function can be graphed and the other half can be determined using symmetry. The graph of an even function is symmetric with respect to the yy-axis. The graph of an odd function is symmetric with respect to the origin.

Even Functions Odd Functions
The yy-axis is a line of symmetry for the graph. The graph has rotational symmetry about the origin, meaning that the graph looks the same after it is rotated 180° about the origin.
The output of the function does not change if xx is replaced with x-x. Replacing xx with x-x changes the sign of the output.
f(x)=f(x)f(-x)=f(x)
f(x)=f(x)f(-x)=-f(x)

Increasing, Decreasing, and Constant Functions

Determining whether a graph is increasing, decreasing, or constant depends on how the xx- and yy-values increase, decrease, or remain the same.

A graph can be increasing, decreasing, or constant over a given interval. In an increasing function, the yy-values increase as the xx-values increase. The graph is going up from left to right. An example of an increasing function is a line with a positive slope.

In a decreasing function, the yy-values decrease as the xx-values increase. The graph is going down from left to right. An example of a decreasing function is a line with a negative slope.

In a constant function, the output values are all the same. The graph is not going up or down. A constant function is a line with a slope of zero.

Knowing whether a graph is increasing, decreasing, or constant can help determine information about the function. For example, if a function represents the profit of a company, the intervals where the graph is increasing or decreasing tell what values will have a profit that is increasing or decreasing.

Increasing Functions Decreasing Functions Constant Functions
The yy-values increase as the xx-values increase. The yy-values decrease as the xx-values increase. The output values are all the same. The graph is a horizontal line.

Maxima and Minima

A function's maxima and minima are determined by the lowest and highest points at specific intervals, called local minimum and local maximum, as well as the function's highest point, called the global maximum, and its lowest point, called the global minimum.

A turning point on a graph is a point where the graph changes from increasing to decreasing or from decreasing to increasing. If the graph changes from increasing to decreasing ( \nearrow\ \searrow), the turning point is a local maximum. If the graph changes from decreasing to increasing ( \searrow\ \nearrow), the turning point is a local minimum. The point where the value of a function is greatest is the global maximum, and the point where the value is least is the global minimum. The maximum value of a function is the greatest value in the range. The minimum value of a function is the least value in the range. If the range of a function extends to infinity, it does not have a global maximum. If the range extends to negative infinity, it does not have a global minimum.

  • A function has a local maximum at x=cx=c if the point (c,f(c))(c, f(c)) on the graph is higher than the points on either side of cc. It is the global maximum if there are no other points on the graph that are higher than f(c)f(c). In this case, f(c)f(c) is the maximum value of ff.
  • A function has a local minimum at x=cx=c if the point (c,f(c))(c, f(c)) on the graph is lower than the points on either side of cc. It is the global minimum if there are no other points on the graph that are lower than f(c)f(c). In this case, f(c)f(c) is the minimum value of ff.
Step-By-Step Example
Analyzing the Graph of a Function
Use the graph to analyze the properties of the function. Assume all relevant information is shown in the graph.
Step 1

Determine the domain and range of the function.

The arrows at the ends of the graph indicate that the graph continues to negative infinity on the left and positive infinity on the right. So the domain of the function is (,)\left(-\infty,\infty\right).

The lowest point on the graph has a yy-value of zero. The arrows are extending up, so the range extends to infinity in the positive direction. So the range of the function is [0,)\left[0,\infty\right).
Step 2

Determine whether the function is even, odd, or neither.

The yy-axis is a line of symmetry for the graph. So, the function is even.
Step 3

Determine where the function is increasing and decreasing.

The graph goes down, then up, then down, and then up. Use interval notation to represent where the xx-values are increasing and decreasing:

  • The function is decreasing on the intervals (,2)(-\infty,-2) and (0,2)\left(0,2\right).
  • The function is increasing on the intervals (2,0)\left(-2,0\right) and (2,)\left(2,\infty\right).
Step 4

Determine the maximum and minimum of the function. The graph has three turning points: (2,0)(-2,0), (0,16)(0,16), and (2,0)(2,0).

The turning points at x=2x=-2 and x=2x=2 are local minima because the graph changes from decreasing to increasing. They are also the lowest points on the graph. So, x=2x=-2 and x=2x=2 are also global minima, while zero is the minimum value of the function.

The turning point at x=0x=0 is a local maximum because the graph changes from increasing to decreasing. However, there are other points on the graph that have a greater yy-value than 16. There is no greatest value in the range. So, the function has no global maximum.
Solution

Summarize the information about the function.

  • Domain: (,)\left(-\infty,\infty\right)
  • Range: [0,)\left[0,\infty\right)
  • Even function
  • Decreasing over (,2)(-\infty,-2) and (0,2)\left(0, 2\right)
  • Increasing over (2,0)\left(-2,0\right) and (2,)\left(2,\infty\right)
  • Local maximum: x=0x=0
  • Global maximum: None
  • Maximum value: None
  • Minimum value: Zero
  • Local minima: x=2x=-2
  • Global minima: x=2x=2