**Definition**

ยค Each input has only one output!

**Example 1**

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

Solution

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**

Words

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

**Hint**

input = number of transcripts

output = price

**Solution**

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

Table

**Solution**

Graph

Solution

Should we connect the dots?

No! We cannot have partial transcripts.

Graphically, is this a function?

Yes! We can use the vertical line test .

**Algebra**

(a)

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

(b)

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?

Does each input have only one output? Yes!

(b) Is it a function?

No! -2 goes to 5 and 0.

**Example 4**

y^{2} - 4 x^{3} + 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

output.

## Function Notation

**Example 1: A quadratic function**

Given f (x) = x^{2} + 3 x, find f (0), f (1), f (x + h),
and .
Solution

**Example 2: A rational function **

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

**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 .

Solution

(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

Solution

**Domain & Range**

** Terminology **

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

**Graphical**

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.

**Algebraic**

**Example 1**

Find the domain and range of
**Solution**

We need a non negative radicand .

domain : x ≥ 0

range : y ≥ 0

**Example 2**

Find the domain of
**Solution**

x - 6 ≥ 0

x ≥ 6

**Example 3**

Find the domain of
**Solution**

x ≠ -3

(No division by 0.)

**Example 4**

Find the domain of
**Solution**

x ∈R

(all real numbers)