Chapter 1.2

Chapter 1.2 focuses on graphs of functions. The topics that we’ll discuss are Graphs of Functions, Increasing and Decreasing Functions, Relative Minimum/Maximum, Step and Piecewise-Defined Functions, and Even/Odd Functions.

Graphs of Functions

A graph of a function is all the combined points of (x, f(x)) plotted on a graph. X is the input of the function and f(x) is the output.

Consider the following function:

 We can find out the domain and range of a function by looking at its graph.

In this example, the domain and range are both  

, as any value of x corresponds to a value of f(x) and vice-versa.

This isn't always the case, however. Consider this next function:

Also, look at its graph.

From this graph, you can see that the domain excludes zero, and that the range excludes all negative values and zero.

Increasing and Decreasing Functions

A piece of a graph of a function can be described as increasing, decreasing, or remaining constant.

A piece of a graph is considered increasing if, as x increases, f(x) increases.
 If, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).

A piece of a graph is considered decreasing if, as x increases, f(x) decreases.
  If, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).

A piece of a graph is considered remaining constant if, as x increases, f(x) doesn't change.

 If, for any x1 and x2 in the interval, x1 < x2 implies f(x1) = f(x2).

Relative Minimum/Maximum

A relative maximum is a point in a graph within two x values that has the highest value for f(x). Said in another way, if you make two imaginary, invisible, vertical lines in a graph, the highest point in between those is the relative maximum. Similarly, a relative minimum is a point in a graph within two x values that has the lowest value for f(x).

Formerly speaking, a function value f(a) is a relative minimum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) <= f(x).
A function value f(a) is a relative maximum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) >= f(x).

Step and Piecewise-Defined Functions

A step function is a noncontinuous function that remains constant for a defined amount of x values, then jumps suddenly to a new y value. Step functions often resemble staircases.

(graph from mathwords.com)

Piecewise-defined functions are functions that have multiple components, depending on the value of x.

Consider the following piecewise-defined function.

Now look at its graph.

Even and Odd Functions

A function is considered even if the graph of it has symmetry with respect to the y-axis (f(-x) = f(x)).

A function is considered odd if the graph of it has symmetry with respect to the origin (f(-x) = -f(x)).

Consider the following function.

f(-x) = (-x)^2

f(-x) = x^2

f(-x) = f(x)

Therefore, this is an even function.

You can also tell this is an even function by looking at its graph.

As you can see, it is reflected across the y axis, so it is an even function.