Algebra Review VII: Functions

A. Determining If a Relation Is a Function

A function is a relationship between two sets. Those two sets are called the domain and range. The domain can be thought of as the set of allowable input values and is generally associated with the variable . The range can be thought of as the set of output values and is generally associated with the variable .

A relationship defined between the two sets is a function if every value of is assigned to only one value of . Given a graph, there is an easy way to tell if it represents a function— it’s called the vertical line test.

Vertical Line Test

If you can draw a vertical line that cuts through the graph in more than one place, the graph cannot represent a function; otherwise, it can.

Example 1. Given the graphs below, which ones represent functions?

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Graph (a) does not represent a function. It is possible to draw a vertical line that cuts the graph in two places. Graphs (b) and (c) do represent functions. For either of these graphs, any vertical line drawn will intersect the graph in at most one place.

Practice Exercises

Use the vertical line test to determine whether the graphs in Practice Exercises 1–4 represent functions.

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B. Function Notation

The most common notation used when identifying a function is .

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This notation is read as “ evaluated at ” or “ of .” Note: In function notation, multiplication is not implied by , as it would be in or .

As an example, let’s look at the squaring function, which takes a number and squares it. The squaring function can be written as . In this function, would be the input and the square of would be the output, so if 2 is in our domain, then 4 would be in our range.

Example 1. Given the squaring function , evaluate and .

For a function, each input can be assigned to at most one output. However, from Example 1 we see that it is possible for more than one input to be assigned to the same output.

When we graph a function , our points are of the form . For the squaring function, the points are of the form .

Example 2. Specify four points on the graph of and sketch the graph.

From Example 1, we know that and (3, 9) are points on the graph. Two more points are and . A graph of the function is shown below.

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Another important function is the identity function: . In the identify function, a number is assigned to itself. So, with an input of , the output is also . Some points on the graph of the identity function are (0, 0), (1, 1), and (2, 2).

Example 3. Graph the identity function.

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The identity function is an example of a linear function, a function of the form . (For the identity function, and .) In this book, you will also find exponential functions, which have the form .

Calculator Note: After entering a function into a TI-84 graphing calculator, you can graph the function and also evaluate the function at specific input values.

Here’s how to graph a function such as using a TI-84 graphing calculator:

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You can also use your calculator to evaluate a function for a specific input value. We continue with the function , which we have already stored in a graphing calculator as Y1. Next, we use the calculator to evaluate . Keep in mind that the calculator knows this function as Y1 instead of as . So, for the calculator you want to determine :

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Practice Exercises

  1. The cubing function can be expressed as . Evaluate and f(2).
  2. What is the form of the points for the cubing function?
  3. Sketch the graph of the function on the -interval from −3 to 3.
  4. is an example of a quadratic function. Evaluate and .
  5. What is the form of the points for ?
  6. Sketch the graph of .
  7. Given , evaluate and .
  8. Given , evaluate and .