function

Objectives
Define Functions.
What are its Types?
What are its Conditions?
Explain Domain.
Explain range.
Use Linear equation in figure.
Use Quadratic equation in figure.

What are Functions?
A relation is a set of ordered pairs. For Eg. the set {(1,3), (4,7), (9,12)} is a
relation.
A set of all first components of the ordered pairs is called the Domain of the
relation.
A set of all second components is Range of the relation.
For the above Relation, the Domain is {1, 4, 9} and Range is {3, 7, 12}
A function is a relation in which no two ordered pairs have the same first
components. It means that a function is a relation for which each member
of the domain is assigned one and only one member of the range.

Functions
If two variables x and y are so related that whenever a value is assigned to x,
one value of y is determined then y is said to be a function of x.
Eg. If a certain fruit sells SR. 100 per Kg, the no.of Kgs bought determines
the price the purchaser pays. For 2 Kgs the price is SR. 200, for three Kgs it
is SR. 300 and so on. Hence, the no.of Kgs and the price paid are
functionally related.

A customary way to designate y as a function of x is,
Y = f(x) which reads, y equals a function of x.
The variable x is called the independent component and the variable y is the
dependent variable
The idea of function includes a domain, range and rule. The rule is usually
given by an equation. The equation for the Eg can be Y = 100x
Functions are used in a wide range of fields including medicine. Example
Medicine doses are calculated on people's masses. A small child will
receive less than a heavier adult.
For a particular medicine the dosage for a 10kg child is 50 millilitres.
Write the equation.

Types of function.
The functions can be classified into following on the basis of variables used :
1. Linear function.
2. Quadratic function
3. Composition function

Linear Function.
A linear function is a function whose graph is a straight
line.
A linear function can be described by a linear equation.
A linear equation is a degree-1 polynomial. In other
words, each term in a linear equation is either a
constant or the product of a constant and a single
variable.
A linear function has the following form
y = f(x) = a + bx

A linear function has one independent variable
and one dependent variable. The independent
variable is x and the dependent variable is y.
In order to be a linear function, a graph must be
both linear (a straight line) and a function
(matching each x-value to only one y-value). A
typical use of a linear function is to convert from
one set of units to another.

Graphing a linear function
To graph a linear function:
1. Find 2 points which satisfy the equation
2. Plot them
3. Connect the points with a straight line
Example:
y = 25 + 5xl let x = 1 then y = 25 + 5(1) = 30 let x = 3 then y = 25 +
5(3) = 40

A quadratic function is a function that can be written in the standard form f(𝓍) = a𝑥
2 + b𝑥 + c, where a ≠ 0. The domain of a quadratic function is all real numbers.
Standard Form of a Quadratic Function
f( 𝑥) = a 𝑥2 + b 𝑥 + c
↓ ↓ ↓
quadratic term linear term constant term
The condition a ≠ 0 gives every quadratic function a quadratic term, but not
necessarily a linear term or a constant term. If a = 0, then the function has no
quadratic term, and it is not a quadratic function.
Quadratic Function

The graph of a quadratic
function is a parabola. The axis
of symmetry is the line that
divides a parabola into two parts
that are mirror images. The
vertex of a parabola is the
point at which the parabola
intersects the axis of symmetry.

The y-value of the vertex of a parabola represents the maximum or minimum
value of the function.

The axis of symmetry of the graph of a quadratic function is always a
vertical line defined by the 𝑥-coordinate of the vertex. In each graph
above, the axis of symmetry is the y-axis, 𝑥 = 0.
Each point of the parabola has a corresponding point on its mirror image.
Two corresponding points are the same distance from the axis of
symmetry. In general, the domain of a quadratic function is all real
numbers.

Composition Function
We can build up complicated functions from simple functions by using the
process of composition, where the output of one function becomes the input
of another

The composition of functions f and g is written f ° g, and is read aloud "f composed with
g." The formula for f ° g is written (f ° g)(x). This is read aloud "f composed with g of x.
(f º g)(x) = f(g(x))
(g º f)(x) = g(f(x))

Example: f(x) = 2x+3 and g(x) =2x
(f º g)(x) = f(g(x)) (g º f)(x) = g(f(x))
f(2x)=2(2x)+3 g(2x+3)=2(2x+3)
f(2x)=4x+3 g(2x+3)=4x+4

Domain of Composition Function
1- Domain of the inside
2- evaluate and simplify
3- Domain of step 2
4- Union

Condition of Function
Examining Ordered Pairs:
An ordered pair is a point on an x-y coordinate graph with an x and y-value.
For example,
(2, -2) is an ordered pair with 2 as the x-value and -2 as the y-value. When given a set of
ordered pairs, ensure that no x-value has more than one y-value paired to it. When
given the set of ordered pairs [(2, -2), (4, -5), (6, -8), (2, 0)], you know that this is not a
function because an x-value -- in this case -- 2, has more than one y-value. However, this
set of ordered pairs [(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)] is a function because a y-value is
allowed to have more than one corresponding x-value.

Solving for Y:
It is relatively easy to determine whether an equation is a function by solving
for y.
When you are given an equation and a specific value for x, there should only
be one corresponding y-value for that x-value.
For example, y = x + 1 is a function because y will always be one greater than
x.
Equations with exponents can also be functions. For example, y = x2 - 1 is a
function; although x-values of 1 and -1 give the same y-value (0), that is the
only possible y-value for each of those x-values.
However, y2 = x + 5 is not a function; if you assume that x = 4, then y2 = 4 + 5 =
9. y2 = 9 has two possible answers (3 and -3).

THE VERTICAL LINE TEST
A test use to determine if a
relation is a function. A
relation is a function if there
are no vertical lines that
intersect the graph at more
than one point.

HORIZONTAL LINE TEST
A test use to determine if a function is
one-to-one. If a horizontal line intersects
a function's graph more than once, then
the function is not one-to-one.
Note: The function y = f(x) is a function if
it passes the vertical line test. It is a one-
to-one function if it passes both the
vertical line test and the horizontal line
test.

Domain
The possible values for the input, or the independent variable, of a function
are the domain of the function.
The domain of a relation is the set of all inputs,or x-coordinates,of the ordered
pairs.You sometimes find the domain of a relation from its graph.

Example
Maria earns $7 per hour for baby-sitting after school and on Saturday. She
works for no more than 16 hours a week.
A.Identify the independent for this situation.
B. Find reasonable domain values for this situation.
★ The amount Maria earns depends on the number of hours she works in a
week. So the number hours she works is the independent variable.
★ A reasonable domain is from 0 to 16 hours.

Range
The range of a relation is the set of all outputs, or y-coordinates of the
ordered pairs. The possible values of the outputs, or the dependent variable,
are the range of the function.
Example
Write the ordered pairs for the relation
Shown below. Find the range
[(2,4) (3,4.5) (4,7.5) (5,7) (6,5) (6,7.5)]
The range is [4,4.5,5,7,7.5]

The Relation Between Domain and Range
Aside from the graph,another way to show a relation is to use a mapping
diagram,which links elements of the domain with corresponding elements of
the range. Write the elements of the domain in one region and the elements
of the range in another. Draw arrows to show how each element from the
domain is Paired with element from the range.
★ Find domain and range relation from the following numbers
Make a mapping diagram for this relation (-1 ,-2), (3, 6), (-5, -10), (3,2)

Linear equation for figure 1
Writing linear equation using the slope-intercept form:
An equation in the slope-intercept form is written as
y=mx+b
m =the slope of the line
This can be done by calculating the slope between two
known points of the line using the slope formula.
b =the y-intercept.
which is The value where a straight line crosses the Y
axis of a graph.
Y intercept
(20,11)
(10,6)

Solving the linear equation for figure 1
1- Make a table: 3- use the equation:
y=mx+b
y= 0.5x + 1
2 - Find the slope:
(6-1)/(20-10)= 0.5
x y
10 1
20 6

Solving a Quadratic Equation using the Quadratic
Formula
1- Put the equation in standard form first.
2- Identify the coefficients, a, b, and c. Be careful to include negative signs if the bx or c
terms are subtracted.
3- Substitute the values for the coefficients into the quadratic formula.
4- Simplify as much as possible.
5- Use the ± in front of the radical to separate the solution into two values: one in which
the square root is added, and one in which it is subtracted.
6- Simplify both values to get the possible solutions.

COMPOSITION OF FUNCTIONS
Functions can also be combined using composition of function.
In a composition, a function is performed, and then a second function is
performed on the result of the first function. The composition of f and g is
denoted by .

References.
http://www.mathplanet.com/education/algebra-1/formulating-linear-
equations/writing-linear-equations-using-the-slope-intercept-form
AlGEBRA 2 , Teacher’s edition ,Pearson Prentice Hall .
Glencoe Algebra 2, Pearson Prentice Hall. Chapter 7. Pages 384- 385
http://sciencing.com/ways-tell-something-function-8602995.html
ALGEBRA1,Teacher’s edition,Pearson Practice Hall.
http://www.mathwords.com/v/vertical_line_test.htm
http://www.mathwords.com/h/horizontal_line_test.htm
http://www.mathplanet.com/education/algebra-1/formulating-linear-
equations/writing-linear-equations-using-the-slope-intercept-form

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