Function notation by sadiq

FUNCTION NOTATION
By.
Mr.sadiqHussain.
C.C.GHAZI

Learning Objectives
In this section, you will:
• Determine whether a relation represents a function.
• Find the value of a function.
• Determine whether a function is one-to-one.
• Use the vertical line test to identify functions.
• Graph the functions listed in the library of functions
• Few graphs of different functions

Explanation;
A jetliner changes altitude as its distance from
the starting point of a flight increases. The
weight of a growing child increases with time.
In each case, one quantity depends on another.
There is a relationship between the two
quantities that we can describe, analyze, and
use to make predictions. In this lesson, we will
analyze such relationships.
Video of shuttle space will be shown

Determining Whether a Relation
Represents a Function
A relation is a set of ordered pairs.
The set of the first components of each ordered
pair is called the domain.
The set of the second components of each ordered
pair is called the range. Consider the following set of
ordered pairs. The first numbers in each pair are the
first five natural numbers. The second number in each
pair is twice that of the first.
{(1,2),(2,4),(3,6),(4,8),(5,10)}
The domain is {1,2,3,4,5}.
The range is {2,4,6,8,10}.

Note that each value in the domain is
also known as an input value,
or independent variable, and is often
labeled with the lowercase letter x. Each
value in the range is also known as
an output value, or dependent variable,
and is often labeled lowercase letter y.

A function f is a relation that assigns a
single value in the range to each value in
the domain. In other words, no x-values
are repeated. For our example that
relates the first five natural numbers to
numbers double their values, this
relation is a function because each
element in the domain, {1,2,3,4,5},is
paired with exactly one element in the
range, The range is {2, 4, 6, 8, 10}

Note that each value in the domain is
also known as an input value, or
independent variable, and is often
labeled
with the lowercase letter x. Each value in
the range is also known as an output
value, or dependent variable, and is
often labeled lowercase letter y

A function f is a relation that assigns a single value
in the range to each value in the domain. In other
words, no
x-values are repeated. For our example that relates
the first five natural numbers to numbers double
their values, this
relation is a function because each element in the
domain, {1, 2, 3, 4, 5}, is paired with exactly one
element in the range,
{2, 4, 6, 8, 10}
Definition of a function

Function
A function is a relation in which each
possible input value leads to exactly
one output value. We say “the output
is a function of the input.” The input
values make up the domain, and the
output values make up the range.

Determining If Class Grade Rules Are Functions
In a particular math class, the overall
percent grade corresponds to a grade
point average.
Is grade point average a
function of the percent grade?
Is the percent grade a function of the
grade point average?
Table 1 shows a possible
rule for assigning grade points.

For any percent grade earned, there is an
associated grade point average, so the grade point
average is a
function of the percent grade. In other words, if we
input the percent grade, the output is a specific
grade point average.
In the grading system given, there is a range of
percent grades that correspond to the same grade
point average. For
example, students who receive a grade point
average of 3.0 could have a variety of percent
grades ranging from 78 all
the way to 86. Thus, percent grade is not a function
of grade point average
Solution:

Using Function Notation
Once we determine that a relationship is a
function, we need to display and define the
functional relationships so
that we can understand and use them, and
sometimes also so that we can program them
into computers. There are
various ways of representing functions. A
standard function notation is one
representation that facilitates working
with functions.

To represent “height is a function of age,” we start
by identifying the descriptive variables h for height
and a for age.
The letters f, g, and h are often used to represent
functions just as we use x, y, and z to represent
numbers and A, B, and
C to represent sets.
h is f of a; We name the function f ; height is a function
of age.
h = f (a); We use parentheses to indicate the function
input.
f(a) We name the function f ; the expression is read as
“f of a.”

Remember, we can use any letter to
name the function; the notation h(a)
shows us that h depends on a. The value
a must
be put into the function h to get a result.
The parentheses indicate that age is
input into the function; they do not
indicate multiplication.

function notation
The notation y = f(x) defines a function
named f. This is read as“y” is a function
ofx under f.” The letter “x” represents
the input value, or independent
variable. The letter y, or f(x),
represents the output value, or
dependent variable.
Y=f(X).

Example :
Using Function Notation for Days in a Month
Use function notation to represent a function whose input is
the name of a month and output is the number of days in
that month.
Solution The number of days in a month is a function of the
name of the month, so if we name the function f, we
write days = f (month) or d = f (m). The name of the month is
the input to a “rule” that associates a specific number
(the output) with each input.

Analysis:
Note that the inputs to a function do not
have to be numbers; function inputs can be
names of people, labels
of geometric objects, or any other element
that determines some kind of output.
However, most of the functions we will
work with in this book will have numbers as
inputs and outputs.

How To…
Given the formula for a function,
evaluate.
1. Replace the input variable in the
formula with the value provided.
2. Calculate the result.

Example 6:
Evaluating Functions at Specific Values
Evaluate f(x) = x 2 + 3x − 4 at:
a. 2 b. a c. a + h d. f (a + h) − f (a).
h
Solution: Replace the x in the function with each
specified value.
a. Because the input value is a number, 2, we can
use simple algebra to simplify.
f (2) = 22 + 3(2) − 4
= 4 + 6 − 4
= 6 Ans.

b.
In this case, the input value is a letter so we
cannot simplify the answer any further.
f (a) = a2 + 3a − 4.Ans.
c.
With an input value of a + h, we must use the distributive
property.
f (a + h) = (a + h)2 + 3(a + h) − 4
= a2 + 2ah + h2 + 3a + 3h −4. Ans.

d.
In this case, we apply the input values to the function more
than once, and then perform algebraic operations
on the result. We already found that
f (a + h) = a2 + 2ah + h2 + 3a + 3h − 4
and we know that
f(a) = a2 + 3a − 4
Now we combine the results and simplify.
f(a+h)-f(a)=(a2 + 2ah + h2 + 3a + 3h − 4 )-(a2 + 3a − 4 )
h h
= (2ah+h2+3h)
h

Example 8:
Solving Functions with the help of a graph:
Given the function h(p) = p2 + 2p, solve for h(p) = 3.
Solution h(p) = 3
p2 + 2p = 3 Substitute the original function h(p) = p2 + 2p.
p2 + 2p − 3 = 0 Subtract 3 from each side.
(p + 3)(p − 1) = 0 Factor.
If (p + 3)(p − 1) = 0, either (p + 3) = 0 or (p − 1) = 0 (or both of
them equal 0). We will set each factor equal to 0
and solve for p in each case.
(p + 3) = 0, p = −3
(p − 1) = 0, p = 1
This gives us two solutions. The output h(p) = 3 when the
input is either p = 1 or p = −3. We can also verify by
graphing as in Figure 6. The graph verifies that h(1) = h(−3) =
3 and h(4) = 24.

Example ; Finding an Equation of a Function
Express the relationship 2n + 6p = 12 as a function p = f (n), if possible.
Solution To express the relationship in this form, we need to be able to write
the relationship where p is a function
of n, which means writing it as p = [expression involving n].
2n + 6p = 12
6p = 12 − 2n Subtract 2n from both sides.
p =2-1/3 n.
Therefore, p as a function of n is written as
p = f (n) = 2-1/3 n.
Analysis;
It is important to note that not every relationship expressed by
an equation can also be expressed as a function
with a formula.

How To…
Given a function represented by a table, identify
specific output and input values.
1. Find the given input in the row (or column) of
input values.
2. Identify the corresponding output value paired
with that input value.
3. Find the given output values in the row (or
column) of output values, noting every time that
output value appears.
4. Identify the input value(s) corresponding to the
given output value.

Example 1
Evaluating and Solving a Tabular Function
Using Table 1,
a. Evaluate g(3) b. Solve g(n) = 6.
Solution
a. Evaluating g (3) means determining the output value of the
function g for the input value of n = 3. The table
output value corresponding to n = 3 is 7, so g (3) = 7.
Can you try for g(n)=6
Table 1

b. Solving g (n) = 6 means identifying the input
values, n, that produce an output value of 6.
Table 1 shows two
solutions: 2 and 4. When we input 2 into the
function g, our output is 6. When we input 4 into
the function g,
our output is also 6.

Finding Function Values from a Graph
Evaluating a function using a graph also requires
finding the corresponding output value for a given
input value, only
in this case, we find the output value by looking at
the graph. Solving a function equation using a
graph requires finding
all instances of the given output value on the graph
and observing the corresponding input value(s).

Example 12
Reading Function Values from a Graph
Given the graph in Figure 7,
a. Evaluate f (2). b. Solve f (x) = 4.

Solution
a. To evaluate f (2), locate the point on the curve
where x = 2, then read the y-coordinate of that
point. The point
has coordinates (2, 1), so f (2) = 1. See Figure .

b.
To solve f (x) = 4, we find the output value 4 on the
vertical axis. Moving horizontally along the line y = 4, we
locate two points of the curve with output value 4: (−1, 4)
and (3, 4). These points represent the two solutions to
f (x) = 4: −1 or 3. This means f (−1) = 4 and f (3) = 4, or
when the input is −1 or 3, the output is 4. See Figure 9.

one-to-one function
A one-to-one function is a function in which
each output value corresponds to exactly
one input value.

Example 1
Determining Whether a Relationship Is a One-to-One
Function
Is the area of a circle a function of its radius? If yes, is the function one-
to-one?
Solution A circle of radius r has a unique area measure given by A = πr2,
so for any input, r, there is only one output,
A. The area is a function of radius r.
If the function is one-to-one, the output value, the area, must
correspond to a unique input value, the radius. Any area
measure A is given by the formula A = πr2. Because areas and radii are
positive numbers, there is exactly one solution:
So the area of a circle is a one-to-one function of the circle’s radius.

Using the Vertical line Test
The most common graphs name the input value x and the
output value y, and we say y is a function of x, or y = f (x)
when the function is named f. The graph of the function is the
set of all points (x, y) in the plane that satisfies the
equation y = f (x). If the function is defined for only a few input
values, then the graph of the function is only a few
points, where the x-coordinate of each point is an input value
and the y-coordinate of each point is the corresponding
output value. For example, the black dots on the graph in Figure
tell us that f (0) = 2 and f (6) = 1. However, the
set of all points (x, y) satisfying y = f (x) is a curve. The curve
shown includes (0, 2) and (6, 1) because the curve passes
through those points.

GRAPH
1
1 2 4
3 5 6
0
1
2 ? (6,1)

The vertical line test can be used to determine
whether a graph represents a function. If we can draw any
vertical line
that intersects a graph more than once, then the graph does not
define a function because a function has only one output value
for each input value. See Figure

Why does the function-vertical line
test work?
Look at the graph below. Notice that graph
touches the vertical line at 2 and -2 when
it intersects the x axis at 4. Therefore when
x = 4 there are two different y-values
(2 and -2). For any input x, a function can
only have one corresponding y value. So
this function FAILS the vertical line test.

Horizontal Line Test
The vertical line test can used with a the horizontal
line test to determine if the original function has an
inverse function. (one to one)
The horizontal line test works similar to the vertical
line test.
This time you draw a horizontal line, and if the line
touches the original function in more than one place it
fails the horizontal line test, and the inverse of the
function is not a function.
If a graph of a function passes both the vertical line
test and the horizontal line test then the graph is " one
to one" and is written f^ -1(x).

This graph passes the vertical line test so it is a function, but fails
the horizontal line test.
.
Therefore, the graph is not one to one and the inverse of this
graph is not a function.

This graph passes the vertical line test so it is a
function, and passes the horizontal line test.
Therefore, the graph is "one to one," and the inverse of
this graph is also a function.

Why does the Horizontal Line Test
work?
A function will pass the horizontal line test if for each y
value (the range) there is only one x value ( the
domain) which is the definition of a function.
If a function passes the vertical line test, and the
horizontal line test, it is 1 to 1.

Function notation by sadiq

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