MM - Functions Lesson

Functions

In past modules, we have looked at various functions. First, review what a function is; a rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set. Recall the vertical line test; that a vertical line can pass through the graph at most once. Also recall that the domain is all of the possible values for x and the range is all of the possible values for y.

Functions we have studied are polynomial, rational, radical, exponential, logarithmic and trigonometric. We will be using these throughout the module and here we will look at some new functions.

A graph represents a function if and only if no vertical line passes through the graph more than one.

Graph of a Function
Function	Not a Function

Piecewise Functions

A piecewise function is a function that is defined differently for different domain values.

An example is equation image indicator $LaTeX: f\left(x\right)=\begin{Bmatrix} x-2& x\le -1 \\ 1&1< x\le 2 \\ 2x&x>2 \end{Bmatrix}$ $f\left(x\right)=\begin{Bmatrix} x-2& x\le -1 \\ 1&1< x\le 2 \\ 2x&x>2 \end{Bmatrix}$

You can see that the y-values are calculated differently, depending upon where in the domain the x- value falls. In order to evaluate a function, you simply find the correct function definition for the given x-value and then use that to calculate the y-value. To find equation image indicator LaTeX: f(-3) $f(-3)$ , first locate -3 in the domain, in the first section $LaTeX: x\le-1$ $x\le-1$ . Then evaluate the corresponding expression, "x - 2, using -3."

equation image indicator $LaTeX: -3-2=-5\:so\:f\left(-3\right)=-5$ $-3-2=-5\:so\:f\left(-3\right)=-5$

Watch the following videos that will explore piecewise functions.

graph with blue line, red line, green line

Function:

$LaTeX: f\left(x\right)=\begin{cases} \color{blue}x+4, x< -4 \\ \color{red}2x+5, -4\le x\le 1\\ \color{green}-x+3,x> 1 \end{cases}$ $f\left(x\right)=\begin{cases} \color{blue}x+4, x< -4 \\ \color{red}2x+5, -4\le x\le 1\\ \color{green}-x+3,x> 1 \end{cases}$

Step Functions

Another type of function (which could be written as a piecewise function) is a step function. A step function is a greatest integer function; which is defined by the greatest integer less than or equal to the value. The symbol for this function is [ ] or [[ ]]. (There is a least integer function that is also a step function, but we will not work with that kind here.)

Here is an example - f(x) =[[x-2]].

Evaluating these functions often involve fractions or decimals for x values.

equation image indicator $LaTeX: f\left(\frac{1}{2}\right)=\left[\left[\frac{1}{2}-2\right]\right]=\left[\left[-\frac{3}{2}\right]\right]=-2$ $f\left(\frac{1}{2}\right)=\left[\left[\frac{1}{2}-2\right]\right]=\left[\left[-\frac{3}{2}\right]\right]=-2$

Inside the double bracket, simplify normally. When you get to the double bracket, take the greatest integer less than or equal to the value inside. The greatest integer less than or equal to -3/2 is -2.

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