Lesson 17MA 152, Sections 3.1 & 3.3

Lesson 17MA 152, Sections 3.1 & 3.3

There exists correspondences such of a value x to a value y such as the examples below.

For every US citizen there is a social security number
For every driver there is a driver's license number
For every city there is one mayor
For every number there is one square of that number

If, for every quantity, there exists (according to some rule) only one corresponding quantity, we have situations that are discussed in this lesson. Each x (called the independent variable or input value) according to some rule is paired to unique y (called the dependent variable or output value).

For example, the equation leads to the following equation and graph that pairs each x with one and only one y.

-10

2-3

-3-8

-2-3

4-15

A function f is a correspondence between a set of input values x (called the domain) and a set of output values y (called the range), where to each x-value in the domain there corresponds exactly one y-value in the range.

To determine if an equation is a function: Solve the equation for y. If y equals one value, then it is a function. The domain and range can also be found.

Ex 1:Determine if each equation is a function or not. Find the domain and range in interval notation.

Function Notation: If y is a function of x, this can be written as using function notation. This notation means that y corresponds to a value x. The notation f (x) means the same as y! Function notation is also a way to 'name' a function.

Ex 2:If , find the following.

Graphs of Functions: A function can be graphed by determining the set of all ordered pairs (points) where x is in the domain and y is in the range. Because each x can only be paired to one y, the vertical line test can be used to determine if a graph represents a function. If every possible vertical line would intersect the graph only once, then the graph represents a function.

Ex 3:Determine which graphs are functions.

Ex 4:Graph each function. Determine the domain and range.

A linear function has the equation form .

There are several types of application problems that are linear function. Given some data that is known to lead to a linear function, you can find the function using these steps.

Write at least two ordered pairs from the data. (Read the problem carefully in order to determine which number is an x and which is a y. The language used should be similar to 'y is a linear function of x'.)
Find the slope of the function using the points.
If the y-intercept is know (b), write the function in the form
If the y-intercept is not known, you must either solve for b and write the function or use point-slope form and solve for y.

Ex 5:The cost of electricity in a local town is a linear function of the number of kilowatt-hours used. If the cost of 400 kwh is $32 and the cost of 600 kwh is $48, find an equation that represents this function in the form .

Ex 6:Celsius temperatures are a linear function of Fahrenheit temperatures. 212º F is equal to 100ºC and 32ºF is equal to 0ºC. Write a linear equation that describes this relationship in the form .

Ex 7:Roberts's television value is a linear function of numbers age of the TV. When the TV was 1 year old, its value was $900. At age 5, its value was $250. Find a linear equation to show its value as a function of age.

3.3

Constant Function: .

Linear Function: .

Quadratic Function:

There are other polynomial functions of higher degrees. The degree of a polynomial function is the largest power of x that appears in the polynomial.

The graph of a linear equation is a line. The graph of a quadratic function is a parabola. The graphs of other polynomial functions are smooth continuous curves; no corners, cusps, breaks, or holes.

If (symmetry about the y-axis), the function is called an even function. If (symmetry about the origin), the function is called an odd function.

Ex 1:Determine if each function is even, odd, or neither even or odd.

Use these steps to graph a polynomial function (Remember, different scales can be used on each axis. Make sure to mark the scale on an axis, unless each 'box' is 1 unit.):

Find any intercepts of the graph.
Find any symmetries of the graph (determine if it is an even, odd, or neither).
Determine where the graph is above and below the x-axis.
If necessary, plot a few more points.
Draw a smooth continuous curve.

Ex 2:Graph each function using the steps.

A Piecewise-Defined Function is defined by using different equations for different intervals of the domain. There may be breaks or jumps in these functions. If there is a value where a break occurs, the equation must meet the definition of a function.

Ex 3:Graph each piecewise-defined function.

x yx y

x y x y x y

May application problem can be represented using piecewise-defined functions. For example:

A cell phone plans charges $0.05 per minute for the first 5 minutes of a call and $0.04 for each additional minute.

Let m = number of minutes of a call. Write a function to represent the total cost of a call with this plan.

A Greatest Integer Function is a function defined as where the value of y for a given x is the greatest integer that is less than or equal to x. For example:

Because the graph of one of these functions is a series of horizontal lines, it is often called a step function.

Ex 4:Graph the following problem using ordered pairs (t, c), where t = the number of minutes and c = total cost.

An on-line information service charges for connect time at a rate of $0.20 per minute, computed for every minute or fraction of a minute.

1 2 3 4 5 6 7

Use the graph to figure the cost of 5 ¼ minutes.