# Math Functions

Functions

• Linear
• Piecewise
• Step
• Exponential

• LINEAR FUNCTIONS- are functions that have x as the input variable, and x is raised only to the first power.

Linear functions come in three main forms:

• Slope-Intercept Form

• Point-Slope Form

• General Form

This linear function:

f(x) = mx + b

May be graphed on the x, y plane as this equation:

y = mx + b

This equation is called the slope-intercept form for a line. The graph of this equation is a straight line. The slope of the line is m. The line crosses the y-axis at b. The point where the line crosses the y-axis is called the y-intercept. The x, y coordinates for the y-intercept are (0, b).

How do you graph linear functions?

Example:

To graph a function such as f(x) = y = 2x + 7, we can select a table of values, plot and connect the points. To fill the table, we choose values for x and use the function to get the values of y as shown in the following table.

 x y -3.5 0 -2 3 -1 5 0 7 0.5 8 1 9 2 11

The graph of the data in the above table is:

The value of y when x is zero in the function is called the y-intercept and the value of x when y is zero is called the x-intercept. For more information on intercepts, please refer to intercepts

The above graph is a linear function of the form y = mx + c where m is the slope of the straight line and c is the y-intercept. The slope of a straight line passing through two points (x1, y1) and (x2, y2) is defined as following: m = ( y2 - y1 )/(x2 - x1)

For the above graph, slope = 2 and the y-intercept = 7.

Example: Plot the graph of a linear function 4x - 3y = 12.

Solution:

To plot the graph of a linear function, usually it is sufficient to plot the x and y intercepts of that linear function.

From the above equation, we find that the x-intercept is 3 (i.e. when y = 0) and the y-intercept is -4 (i.e. when x = 0). For convenience, we will also find another point: (x= 6, y=4). Now the graph can be plotted as following:

• PIECEWISE FUNCTIONS-Some functions are made up of several different rules for different subsets of their domains. Such functions are often referred to as hybrid functions. A special case of hybrid functions is when each rule is a linear function; in this case we have what is known as a piecewise functions. An example of such an example is F(x) = {x - 2 if x > 4 , 2x - 6 if x < 4 To sketch this graph we simply sketch the graph of y = x – 2 for x > 2 and then, sketch the graph of y = 2x – 6 for x < 4

a more complez form of a piecewise function is as follows

the graph of the above looks like this:

• STEP FUNCTIONS- Another type of piecewise linear function is the greatest integer functions (y = [x]), which is also known as the step function. These graphs are made up of discontinuous horizontal lines (segments) that end up resembling a staircase. Such graphs are useful when modeling the cost of mailing parcels of different weights. For example, if it costs \$3.00 to send parcels weighing between at least 1 kg but less than 3 kg, \$7.00 for parcels weighing at least 3 kg but less than 5 kg, \$13.00 for parcels weighing at least 5 kg but less than 8 kg and so on. *piecewise and step function instructions are courtesy of "Mathematical Studies" IBID Press, Victoria, 1998.

a step function looks like this:

• EXPONENTIAL FUNCTIONS- Exponential functions are shown in the basic form: f(x)=aⁿ . These are defined for rational values of “x” by using the algebraic definitions in terms of repeated multiplication and laws of exponents.

some major exponential rules include:

• Rule 1: To multiply identical bases, add the exponents.

Example: means which in turn can be written . According to Rule 1, you can get to the answer directly by adding the exponents:

• Rule 2: To divide identical bases, subtract the exponents.

Example: can be written as: .

• Rule 3: When there are two or more exponents and only one base, multiply the exponents.

Example: According to Rule 3, we can go directly to the answer by multiplying the exponents...

.

For a > 0 the value of is always positive, no matter what x is. For every positive the exponential function with base a is one-to-one. It is increasing if a > 1 with a horizontal asymptote at y = 0 on the left, i.e. the graph approaches the x-axis as x->- and decreasing with a horizontal asymptote at y = 0 on the right if a < 1.

For example, if the base is 2 and x = 4, the function value f(4) will equal 16. A corresponding point on the graph of would be (4, 16):

The general behaviour of more complicated functions involving exponentials can often be figured out on the basis of whether the powers involved have positive or negative exponents. The exponent is always negative and becomes even more negative as . So the value of is always less than or equal to1 and the graph has a horizontal asymptote at y = 0 on both sides.

Logarithmic Functions are also related to exponential functions. Logarithmic functions are the inverse of exponential functions. The exponential equation could be written in terms of a logarithmic equation as . The exponential equation can be written as the logarithmic equation .

The graphs are similar too, and since logarithms are nothing more than exponents, you can use the rules of exponents with logarithms.

For example: if (4, 16) is a point on the graph of an exponential function, then (16, 4) would be the corresponding point on the graph of the inverse logarithmic function: