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)=x−3x−1
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 R−1
Now let’s find the range using the inverse function method
y=x−3x−1
y(x−1)=x−3
xy−y=x−3
3−y=x−xy
x=3−y1−y
It is very clear that x assumes real values for all y except y=1, So Range is
R–1
b. f(x)=x2−16x−4
First let’s see the domain
We can see that function is defined for all values of x except 4
So Range is R−4
Now f(x)=x2−16x−4
f(x)=(x−4)(x+4)x−4
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 x≠4, so y=8 would not be in the range of the function.
So,Range is R–4
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)=x2−1
Clearly, it is defined for all values of x, Domain =R
Now
y=x2−1
x2=y+1
x=±√y+1
For x to be real , y+1≥0 or y≥−1
So Range=[−1,∞)
b. f(x)=−2x2−1
Clearly, it is defined for all values of x, Domain =R
Now
y=−2x2−1
y+1=−2x2
x2=−1−y2
x=±√−1−y2
For x to be real , −1−y2≥0
or −1−y≥0
−1≥y
y≤−1
So Range=(∞,−1]
How to find the range of a modulus function
A modulus Function is like f(x)=|x−1|
a. f(x)=1–|x−3|
Clearly, it is defined for all values of x, Domain =R
Now
y=1−|x−3|
|x−3|=1−y
Clearly for real values of x, 1−y≥0
or
1≥y
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)=x2–2x+2
b. f(x)=ax−bcx−d
c. f(x)=1√x−6
d. f(x)=x2–5x+6
e. f(x)=x−25−x
I hope you like this article on how to find the range of a rational function
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