MATHEMATICS 1MATHEMATICS 1MKK / TM1 1101 / 3 sks

Dr. Fauzun

ISI KULIAH 2:2. Functionsa. Definitionb. Variablec. Macam-macam Function dan Grafik

− Linear− Polynomial− Exponential− Logaritmic− Hyperbolic− Trigonometry

     

A function is an operation performed on an input (x) to produce an output (y = f(x) ).

 The Domain of f is the set of all allowable inputs (x values)The Range of f is the set of all outputs (y values)

 

f

x y =f(x)

Domain Range

 

Functions

What is a Variable?

• Simply, something that varies.• Specifically, variables represent persons or objects

or anything that can be manipulated, controlled, or merely measured for the sake of research.

• Variation: How much a variable varies. Those without (sometime with little) variation are called constants.

Commonly, x and y are called variable

There are two kind of variables:

1.Independent variable:

2.Dependent variable:

Variables that you put into the equations

Variables that you solve for

Example:

y = 2x - 4

Using the equation above, y has to be the variable

Why? Because it is the variable that is being solved for

Dependent

For above example, because linear equation, it has two variables (x & y), there are many possible combinations of answers

Those possible solutions are written as ordered pairs. Usually an ordered pair is written with the independent variable first then the dependent variable >>>>> (x,y)

Most of the time when we use the variables x & y in an equation, x is independent and y is dependent

There are some exceptions, but we will work with those at a much later time

As said before, there are many possible ordered pairs

To find more than one solution, we use a Table of Values

x y2 3-2 10 -3.... ....

To be well defined a function must

−Have a value for each x in the domain−Have only one value for each x in the domain  e.g: y = f(x) = √(x-1), x is not well defined as if x < 1 we will be trying to square root a negative number.

y = f(x) = 1/(x-2), x is not well defined as if x = 2 we will be trying to divide by zero.

This is not a function as some x values correspond to two y values.

Domain

y = (x-2)2 +32

The Range is f(x) ≥ 3

Finding the Range of a function Draw a graph of the function for its given DomainThe Range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph.

Range

Domain

3

y = (x-2)2 +3

The Function is f(x) = (x-2)2 +3 , x

The Function is f(x) = 3 – 2x , x

The Range is f(x) < 3

fx g(f(x))

= gf(x)

gf(x)

Finding gf(x)

Note: gf(x) does not mean g(x) times f(x).

Note : When finding f(g(x))Replace all the x’s in the rule for the f funcion with the expression for g(x) in a bracket.e.g If f(x) = x2 –2xthen f(x-2) = (x-2)2 – 2(x-2)

Composite Functions

      

 

gf(x) means “g of f of x” i.e g(f(x)) .First we apply the f function.Then the output of the f function becomes the input for the g function.Notice that gf means f first and then g.

Example if f(x) = x + 3, x and g(x) = x2 , x thengf(x) = g(f(x)) = g(x + 3) = (x+3)2 , xfg(x) = f(g(x)) = f(x2) = x2 +3, xg2(x) means g(g(x)) = g(x2) = (x2)2 = x4 , xf2(x) means f(f(x)) = f(x+3) = (x+3) + 3 = x + 6 , x

Notice that fg and gf are not the same. The Domain of gf is the same as the Domain of f since f is the first function to be applied.The Domain of fg is the same as the Domain of g. For gf to be properly defined the Range (output set) of f must fit inside the Domain (input set) of g.

For example: if g(x) = √x , x ≥ 0 and f(x) = x – 2, xThen gf would not be well defined as the output of f could be a negative number and this is not allowed as an input for g.However fg is well defined, fg(x) = √x – 2, x ≥ 0.

Domain of f Range

of fb

f

a

f-1

= Domain of f-1

= Range of f-1

Note: f-1(x) does not mean 1/f(x).

Inverse Functions.

 The inverse of a function f is denoted by f-

1 .The inverse reverses the original function.

So if f(a) = b then f-1(b) = a 

 

 

One to one Functions If a function is to have an inverse which is also a function then it must be one to one.This means that a horizontal line will never cut the graph more than once.i.e we cannot have f(a) = f(b) if a ≠ b,Two different inputs (x values) are not allowed to give the same output (y value).For instance f(-2) = f(2) = 4

y = f(x) = x2 with domain x is not one to one. So the inverse of 4 would have two possibilities : -2 or 2.This means that the inverse is not a function.

We say that the inverse function of f does not exist.If the Domain is restricted to x ≥ 0Then the function would be one to one and its inverse would be f-1(x) = √x , x ≥ 0

Domain

The domain of the inverse = the Range of the original.So draw a graph of y = f(x) and use it to find the Range

Finding the Rule and Domain of an inverse function

       

Rule

Swap over x and yMake y the subject

Drawing the graph of the Inverse The graph of y = f-1(x) is the reflection in y = x of the graph of y = f(x).

Example:Find the inverse of the function y = f(x) = (x-2)2 + 3 , x ≥ 2 Sketch the graphs of y = f(x) and y = f-1(x) on the same axes showing the relationship between them. DomainThis is the function we considered earlier except that its domain has been restricted to x ≥ 2 in order to make it one-to-one.We know that the Range of f is y ≥ 3 and so the domain of f-1 will be x ≥ 3.

Note: we could also have -√(x –3) = y-2and y = 2 - √(x –3)But this would not fit our function as y must be greater than 2 (see graph)

RuleSwap x and y to get x = (y-2)2 + 3Now make y the subjectx – 3 = (y-2)2

√(x –3) = y-2y = 2 + √(x –3) So Final Answer is:f-1(x) = 2 + √(x –3) , x ≥ 3

GraphsReflect in y = x to get the graph of the inverse function.

Note: Remember with inverse functions everything swaps over.Input and output (x and y) swap overDomain and Range swap overReflecting in y = x swaps over the coordinates of a point so (a,b) on one graph becomes (b,a) on the other.

Macam-macam Functions dan Grafik

o Linear

o Polynomial

o Exponential

o Logaritmic

o Hyperbolic

o Trigonometry