Question

How to find the range of a rational function algebraically?

Answer 1

Domain and Range of Rational Functions

The  domain  of a  function f(x) is the set of all values for which the function is defined, and the  range  of the function is the set of all values that f takes.

A rational function is a function of the formf(x)=p(x)q(x) , where p(x) and  q(x) are polynomials and q(x)≠0 .

The domain of a rational function consists of all the real numbers x except those for which the denominator is 0 . To find these x values to be excluded from the domain of a rational function, equate the denominator to zero and solve for x .

For example, the domain of the parent function f(x)=1x  is the set of all real numbers except x=0 . Or the domain of the function f(x)=1x−4  is the set of all real numbers except x=4 .

Now, consider the function f(x)=(x+1)(x−2)(x−2). On simplification, when x≠2 it becomes a linear function f(x)=x+1 . But the original function is not defined at x=2 . This leaves the graph with a hole when x=2 .



One way of finding the range of a rational function is by finding the domain of the inverse function.

Another way is to sketch the graph and identify the range.

Let us again consider the parent functionf(x)=1x . We know that the function is not defined when x=0 .

As x→0  from either side of zero, f(x)→∞ . Similarly, as x→±∞,f(x)→0 .



The graph approaches x -axis as x tends to positive or negative infinity, but never touches the x -axis. That is, the function can take all the real values except 0 .

So, the range of the function is the set of real numbers except 0 .
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Domain and Range of Rational Functions

The  domain  of a  function f(x)fx is the set of all values for which the function is defined, and the  range  of the function is the set of all values that ff takes.

A rational function is a function of the formf(x)=p(x)q(x)fx=pxqx , where p(x)px and  q(x)qx are polynomials and q(x)≠0qx≠0 .

The domain of a rational function consists of all the real numbers xx except those for which the denominator is 00 . To find these xxvalues to be excluded from the domain of a rational function, equate the denominator to zero and solve for xx .

For example, the domain of the parent function f(x)=1xfx=1x  is the set of all real numbers except x=0x=0 . Or the domain of the function f(x)=1x−4fx=1x−4  is the set of all real numbers except x=4x=4 .

Now, consider the functionf(x)=(x+1)(x−2)(x−2)fx=x+1x−2x−2 . On simplification, whenx≠2x≠2 it becomes a linear functionf(x)=x+1fx=x+1 . But the original function is not defined at x=2x=2 . This leaves the graph with a hole when x=2x=2 .



One way of finding the range of a rational function is by finding the domain of the inverse function.

Another way is to sketch the graph and identify the range.

Let us again consider the parent functionf(x)=1xfx=1x . We know that the function is not defined when x=0x=0 .

As x→0x→0  from either side of zero,f(x)→∞fx→∞ . Similarly, asx→±∞,f(x)→0x→±∞,fx→0 .



The graph approaches xx -axis as xx tends to positive or negative infinity, but never touches the xx -axis. That is, the function can take all the real values except 00 .

So, the range of the function is the set of real numbers except 00 .

Example 1:

Find the domain and range of the functiony=1x+3−5y=1x+3−5 .

To find the excluded value in the domain of the function, equate the denominator to zero and solve for xx .

x+3=0⇒x=−3x+3=0⇒x=−3

So, the domain of the function is set of real numbers except −3−3 .

The range of the function is same as the domain of the inverse function. So, to find the range define the inverse of the function.

Interchange the xx and yy .

x=1y+3−5x=1y+3−5

Solving for yy you get,

x+5=1y+3⇒y+3=1x+5                          ⇒y=1x+5−3x+5=1y+3⇒y+3=1x+5                          ⇒y=1x+5−3

So, the inverse function isf−1(x)=1x+5−3f−1x=1x+5−3 .

The excluded value in the domain of the inverse function can be determined byequating the denominator to zero and solving for xx .

x+5=0⇒x=−5x+5=0⇒x=−5

So, the domain of the inverse function is the set of real numbers except −5−5 . That is, the range of given function is the set of real numbers except −5−5 .

Therefore, the domain of the given function is {x∈R  |  x≠−3}{x∈ℝ  |  x≠−3}  and the range is {y∈R  |  y≠−5}{y∈ℝ  |  y≠−5} .