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how to find the range of a function algebraically

how to find the range of a function algebraically, 	how to find range of a rational function

For a function
f:A ->B

Set A is called the domain of the function f
Set B is called the co-domain of the function
The set of Images of all elements in Set A is called the range i.e it is the set of values of f(x) that we get for each and every x in the domain

Now let’s see how to find the range of a function algebraically i.e without plotting the graph

How to find the range of a function algebraically

General Method is explained below. This is called the inverse function technique

(a) put y=f(x)
(b) Solve the equation y=f(x) for x in terms of y ,let x =g(y)
(c) Find the range of values of y for which the value x obtained are real and are in the domain of f
(d) The range of values obtained for y is the Range of the function

This is basically how to find the range of a function without graphing

Let’s see fee examples with various types of functions

How to find the range of a rational function

a. f(x)=x3x1

First, let’s see the domain of the function

We can see that function is defined for all values of  x except 1

So Range is R1

Now let’s find the range using the inverse function method

y=x3x1

y(x1)=x3

xyy=x3

3y=xxy

x=3y1y

It is very clear that x assumes real values for all y except y=1, So Range is

R1

b. f(x)=x216x4

First let’s see the domain

We can see that function is defined for all values of  x except 4

So Range is R4

Now f(x)=x216x4

f(x)=(x4)(x+4)x4

f(x)=x+4

or y =x+4

x= y -4

It is very clear that x assumes real values for all y

But there is one catch, we got this equation only when x4, so y=8 would not be in the range of the function.

So,Range is R4

How to find the range of a quadratic function/polynomial function

A quadratic Function /Polynomial function is like f(x)=ax2+bx+c.
a. f(x)=x21

Clearly, it is defined for all values of x, Domain =R

Now

y=x21

x2=y+1

x=±y+1

For x to be real , y+10 or y1

So Range=[1,)

b.  f(x)=2x21

Clearly, it is defined for all values of x, Domain =R

Now

y=2x21

y+1=2x2

x2=1y2

x=±1y2

For x to be real ,  1y20

or 1y0

1y

y1

So Range=(,1]

How to find the range of a modulus function

A modulus Function is like f(x)=|x1|
a. f(x)=1|x3|

Clearly, it is defined for all values of x, Domain =R

Now

y=1|x3|

|x3|=1y

Clearly for real values of x, 1y0

or

1y

So Range is (,1]

Similarly, we find the range of many functions algebraically i.e without plotting the graph. (how to find range of a function without graphing)

Some Practice Questions

a. f(x)=x22x+2

b. f(x)=axbcxd

c. f(x)=1x6

d. f(x)=x25x+6

e. f(x)=x25x

I hope you like this article on how to find the range of a rational function

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