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Function

Definition

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?

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

Complete the fol lowing table .

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

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) 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?

(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?

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
output.

Function Notation

Example 1: A quadratic function

Given f (x) = x2 + 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)

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