Download Lesson 3-2 Functions and Function Notation

Warm Up
1.
2.
3.
Find a triple if r = 10 and s = 2.
If a = 6, b = 8 and the triangle is a
right triangle, what is the value of c?
If c = 50 and b = 48, what is the
value of a if it is a right
triangle?
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Lesson 4 Functions and
Function Notation
Objective: To learn about functions,
domains and ranges. To use function
notation to represent functions
Functions
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


y = 3x + 1
For this equation, every x we
choose, will give us a new y.
x is the input, and y is the output.
For every x we choose there is one
and only one y possible – therefore
this is a FUNCTION
Definition of a Function

A function is a correspondence between
two sets X and Y that assigns to each
element x of set X exactly one element
y of set Y
Domain and Range

For each element x in X, the
corresponding element y in Y is
called the image of the function
at x. The set X is called the
domain of the function, and the
set of all function values, Y is
called the range of the function.
Ex 1: Determine whether each
relation is a function.
a. {(4,5), (6, 7), (8,8)}
b. {(5, 6), (4, 7), (6, 6), (6, 7)}
Solution We begin by making a figure
for each relation that shows set X , the
domain, and set Y , the range.
Solution for part (a)
X
Y
The figure shows that every
4
6
8
5
7
8
element in the domain
Domain Range
corresponds to exactly one
element in the range.
No two ordered pairs in the given relation have
the same first component different second
components. Thus, the relation is a function
Solution for part (b)
X
Y
4
5
6
6
7
The figure shows that 6
corresponds to both 6 and 7.
Domain Range
If any element in the domain corresponds to
more than one element in the range, the
relation is not a function, Thus, the relation
is not a function.
Vertical Line Test (Pencil Test)

Given a graph – any vertical line
drawn on the graph should only go
through one point.
Function
Not a Function
Function Notation
y  f ( x)
The variable x is called the independent
variable, because it can be assigned any
of the permissible numbers from the domain.
The variable y is called the dependent
variable, because its value depends on x.
Function Notation
 The
special notation f(x),
read “f of x”, represents the
value of the function at the
number x.
 The notation does not mean
“ f times x.”
Finding a Function’s Domain

The domain (the x’s) of a function
can be almost any real number.
There are two cases when there are
exceptions:

Division by zero – if x is in the bottom
of the fraction (denominator), then x
cannot make that denominator 0.

f(x) =
1
; x = 3 is not defined.
x 3
Finding a Function’s Domain

Even roots of negative numbers
y =
x  1 is only a real number if x ≥
1
Finding a Function’s Domain
Exclude from a function's domain real
numbers that cause division by zero
and real numbers that result in an even
root of a negative number.
Ex 5: Find the domain of each
function
a. f ( x)  8 x  5 x  2
2
2
b. g ( x) 
x5
c. h( x)  x  2
Solution part a
The function f ( x)  8 x  5 x  2 contains
2
neither division nor an even root.
The domain of f is the set of all real numbers.
Solution part b
2
The function g ( x) 
contains division.
x5
Because division by zero is undefined, we
must exclude from the domain values of x
that cause x  5 to be 0. Thus x cannot equal
to  5. The domain of function g is {x | x  5}.
Solution part c
The function h( x)  x  2 contains an even
root. Because only non-negative numbers
have real square roots, the quantity under the
radical sign, x  2 must be greater than or
equal to 0. Thus, x  2  0 or x  2
Therefore the domain of h is {x | x  2}
or the interval [ 2, ).
Practice Exercises
Find the domain of each function:
12 x
1. h( x)  2
x  36
1
2. f ( x) 
x2
Answers
1. {x | x  6, x  6}
2. {x | x  2} or (  2, )
Graphing Functions

Using a graphing calculator



Put the equation or function into the
form y = ….
Button at top left y =
Then type in the rest of the equation
and press [GRAPH] at top right.
Evaluating Functions
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
Given f(x) = x2 + 2x – 1, find f(2).
This means to plug in 2 wherever
there is an x in the function.

f(2) = (2)2 +2(2) – 1
=4+4–1
=7
Evaluating Functions

Given f(x) = x2 + 2x – 1,
find f(–3).

f(–3) = (–3)2 +2(–3) – 1
=9–6–1
=2
Practice

f(x) = x2 + 3x- 4, find f(-2) & f(0)

f(x) = x2 + 1, find f(-3) & f(x-1)
Practice

Find the numbers for x whose
image is 2.

f(x) = x2