Evaluating Functions

Now that you know how to determine if a relation is a function, let's evaluate some functions. Sometimes you'll hear the term "function machine." This is because in order to evaluate a function, an input goes in, a rule happens, and an output comes out the other end. Here's a visual of the function machine.

input into rule out of output

You could think of this much like a vending machine. You put money in (input/domain), make a drink selection (rule), and you'll get a drink (output/range).

If we have the function f(x)=x+1, then our "rule" is x+1. The value that we choose for x is the input. Once we substitute an input , x, into the rule , function, you'll have an output , or value for y.

To evaluate f(4), read "f of 4," substitute 4 in for x, or the input.

f(x) = x + 1
f(4) = 4 + 1

f(4) = 5

This tells us the ordered pair (4,5) is found on the graph of the function f(x)=x+1.

You can also evaluate a function at a particular input value if you have a graph. Remember, the input and output form an ordered pair found on the graph of the function.

image of graph

The function f(x) is shown on the above graph. Find f(2).

To find f(2), we look for the x value of 2, and determine the output, or y value. The point containing 2 as the x coordinate is (2,0). The y value of that point is 0, making f(2)=0.

image saying "practice your skills" You try some. Using the same graph above, evaluate the following.

f(8)
- Solution: 5
f(-2)
- Solution: -2
f(-6)
- Solution: 2
f(0)
- Solution: 1
f(-9)
- Solution: -1

EXAMPLE:

Consider these tables of values. Determine the rule. Express each function verbally and algebraically.

Input	-4	3	5	16	25	31
Output	-7	7	11	33	51	63

EXAMPLE:

Now, what if you are given the function and you need to create a table of values and graph the function?

Create a table and graph for y = x² + 1

Input
Output

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