How to find Domain and Range for rational functions

Today I will explain about Domain and Range for 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 form f(x)=p(x)/q(x), where p(x) and q(x) are the polynomials and q(x)is not equal to zero. Hence the domain of the rational function consists of all real numbers except those for where the denominator is 0. To find these values to be excluded from the domain of a rational function, equate the denominator to zero and solve for x.

For example, in f(x)=1/x, the domain of the function is the set of all real numbers except 0.

• f(x)=1/x-4 ,the domain of the function is the set of all real numbers except 4
• f(x)=(x-1)/(x-3), clearly, f(x) takes all real values except for the values of where x-3=0 i.e. x=3.Hence domain of f, d(f)= R-{3}

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

We can also sketch the graph and identify the range.

Let us again consider the parent function f(x)=1/x. We know that the function is not defined at x=0. As x tends to 0, then f(x) tends to infinite. Similarly As x tends to infinite, f(x) tends to 0. The graph approaches x axis as x tends to positive or negative infinity, but never touches the x axis.That means the function can takes all real values except 0. So the range of the function is set of real numbers except 0.