Rational Functions Examples

Examples 1: Find the vertical asymptote of (x+1)/(2x+3).

Solution:

Given f(x) = (x+1)/(2x+3)

As the function is already in its lowest form and there is no common factor between numerator and denominator, vertical asymptote is calculated by simply equating the value of denominator equal to zero.

2x + 3 = 0

x = -3/2

Thus, x = -3/2 is the vertical asymptote of the given function.

Example 2: Find the horizontal asymptote of x³+x²+1/(2x+1).

Solution:

Given f(x) = x³+x²+1/(2x+1)

Degree of numerator N = 3

Degree of denominator D = 1

As N>D, there is no horizontal asymptote for given function.

Example 3: Find the oblique asymptote of (2x²+3)/(x+4).

Solution:

Given f(x) = 2x²+3/(x+4)

As degree of numerator = 2 = degree of denominator + 1, oblique asymptote is the quotient that is obtained by dividing numerator and denominator. The division is shown below:

As the quotient obtained is 2x, it is the oblique asymptote to the given function.

Example 4: Find the domain of the function x/(x²-1).

Solution:

Given f(x) = x/(x²-1)

• Equate the denominator to zero.

x² – 1 = 0

x = +1 or -1

Thus domain of the given function is set of real numbers excluding 1 and -1 which is R – {-1, 1}.

Example 5: Find the range of the function 4x+5/(6x+7).

Solution:

Given f(x) = 4x+5/(6x+7)

• Set f(x) = y

y = 4x+5/(6x+7)

• Solve for x

y(4x+5) = 6x+7

4xy+5y = 6x+7

4xy-6x = 7-5y

x(4y-6) = 7-5y

x = 7-5y/(4y-6)

• Keep denominator not equal to zero

4y – 6 ≠ 0

y ≠ 6/4

y ≠ 3/2

• Thus the range of function is set of real numbers excluding 3/2 which is R – {3/2}

Rational Function is a type of function that is expressed as a fraction where both the numerator and denominator must be a polynomial and the denominator can never equal zero. Thus a rational function is similar to a fraction but the numerator and denominator are polynomial functions. In simple words, the rational function can be defined as the ratio of two polynomials. Rational functions find applications in various daily life problems and in various fields in life.

In this article, we shall discuss rational function in detail.