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A function is a rule that assigns each element in a set A exactly Algebra -helper/ exponent -power- zero .html">one element in a Set B.

ยค Each input has only one output!

Example 1

Is the price of a flight a function of the length of the flight?

This is not a function. The same length of flight can have multiple prices (where you buy/when you buy/where you sit/how
much baggage).

Example 2

The registrar charges $5 for each transcript. Is this a function?

input = number of transcripts
output = price

Yes, there is only one price for any given number of transcripts. That is, each input has only one output.


Complete the fol lowing table .

where n is the number of transcripts ordered and c (n) is the cost for n transcripts.


Should we connect the dots?
No! We cannot have partial transcripts.
Graphically, is this a function?
Yes! We can use the vertical line test .


(a) Find an ex pression for c (n) where n is the number of transcripts ordered and c(n) is the cost.
(b) What does c(n) represent? Is c(n) the independent variable or the dependent variable?

The cost will be equal to 5 times the number of transcripts. c(n) = 5 n

c (n) represents the cost, in dollars, of buying n transcripts.
c (n) is the dependent variable.

Domain and Range
Domain : the set of whole numbers {0, 1, 2, 3, ...}
Range : 5n where n is a whole number

Example 3
(a) Is it a function?

x -2 0 2 4
y 5 2 0 2

Does each input have only one output? Yes!

(b) Is it a function?

x -2 1 -2 4
y 5 2 0 3

No! -2 goes to 5 and 0.

Example 4

y2 - 4 x3 + 6 = 0
(a) Is y a function of x?
(b) Is x a function of y?

(a) Solve for y . Does every input have only one output?

Not a function!

(b) Solve for x. Does every y- value have only one x-value?

Why are functions helpful?
Prediction and modeling

Vertical Line Test
If we pass a vertical line across the graph and it touches the graph only once for each vertical line, then each input has only one

Function Notation

Example 1: A quadratic function

Given f (x) = x2 + 3 x, find f (0), f (1), f (x + h), and .


Example 2: A rational function

Given find g(0), g(5), g(-2), and g(6).


More on Function Values
Graphical Manipulate

(a) Find f (2).
(b) Find f (7).
(c) Find x such that f (x) = 7.
(d) Find x such that f (x) = 4 .

(a) f (2) ≈ 4.2
(b) f (7) ≈ 4
(c) x ≈ 8.6
(d) x ≈ 1.8, 4.6, 7

Graphical Example

Example: Find all x such that f (x) = 0.

f (x) = x(x - 5)2 (2 x + 3)
Solve for x.

x(x - 5)2 (2 x + 3) = 0


Domain & Range

A function as signs each input one and only one output.
Input ---------------------------> Output
The domain of a function is the set of all possible inputs. The range of a function is the set of all outputs.
Domain -------------------------> Range
The independent variable represents the input of the function. The dependent variable represents the output of the function.
Independent Variable --------> Dependent Variable


Use the graph of f(x) and g(x) below to answer parts a-f. Note that g(x) is the thicker graph and f(x) is the thinner graph.

(a) Find the domain of f(x).
(b) Find the range of f(x).

The domain of the thinner function, f (x), is (-1, 4]. This can be written as -1 < x ≤ 4.
The range of f (x) is (-3, 4], that is -3 < y ≤ 4.


Example 1
Find the domain and range of

We need a non negative radicand .

domain : x ≥ 0
range : y ≥ 0

Example 2

Find the domain of

x - 6 ≥ 0
x ≥ 6

Example 3

Find the domain of

x ≠ -3
(No division by 0.)

Example 4

Find the domain of

(all real numbers)

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