Unit 1: FunctionsUnit 1: FunctionsUnit 1: FunctionsUnit 1: Functions

1-1: Key Features of Functions1-1: Key Features of Functions

and Real-life Functionsand Real-life Functions

Function Vocabulary

Relation Domain Range Independent variable Dependent variable Function Function notation

Recall:

Ex. 1

Recall: Function Notation

When the relation is determined to be a function, and you can write a rule to describe it, it is written in function notation.

For this domain of {-3, 0 ,¼}, you have a range of {11, 5, 4½}

Ex. 2

You may be asked to graph a function that uses a specific domain.Ex. Graph the function f(x) = 10/x, where the domain is the integers between 1 and 6, inclusive. (inclusive means that the x values of 1 and 6 willbe used, and graphed with closed points).

Ex. 3

The function consists ofthe 6 points that aregraphed. This is calleda discontinuous function because it is not a smoothcurve.

The domain is always along the x-axis. When you are looking at a graph, the interval will be from the smallest point to the largest. It doesn’t matter how high or low the graph is.

Smallest point Largest point

With continuous functions, you will have domain and range intervals

(cont’d) Identifying domain and range from a graph.

The range is always along the x-axis. When you are looking at a graph, the interval will be from the smallest point to the largest. It doesn’t matter where it begins and ends.

Smallest point

Largest point

Here, the domain is written:

The range is written:

This is strictly using inequalities. There is other notation that can be used.

Using a graph to identify the domain and the range.

Write the domain and range, given the graph of the function.

Notice that the graph extends along the x-axis from -5 to 7, and the points are open, which means they are not included (< or >) in the domain. Thus, the domain is: -5 < x < 7The high point of the graph is 8, and the low point of the graph is -6. Here, 8 is Not included, but -6 is (that point is not open, but closed). Thus, the range is:-6 y < 8

Ex. 4

Take what you have learned to draw a function, given the domain and the range.

Example 5

Functions in the Real World

Sometimes, you are not told which variable depends on the other. In this case,You will need to make the decision.

The time it takes you to get home from the football game and the speed you drive are related to each other. Sketch a reasonable graph showing this relationship.

Ex. 6

Ask yourself questions to decide which is the independent variable and which is the dependent. What would those questions be?

Which question sounds more reasonable?

Now you can draw your axes, and label them with time and speed.

When you have enough points, you can connect them with a curve (which can also be a line).

Since it always takes you some time to get home no matter how fast you drive, the graph never touches the horizontal axis.

It never touches thevertical axis either, since you would neverget home if your speed were zero.

An asymptote is a line which a graphgets close to, but never touches

Ex. 7 You take a roast beef out of the refrigerator and put it into a hot oven. The temperature of the beef depends on how long it has been in the oven. Sketch a reasonable graph.