Rational Function: End Behavior of Rational Functions

What is rational function & end behaviour of rational functions? A rational function is a ratio of polynomials where the polynomial in the denominator shouldn’t be equivalent to zero. Isn’t it reaching the definition of a rational number (which is of the form p/q, where q ≠ 0)? Did you know rational functions see the application in different fields in our day-to-day life? Not only do they represent the relationship between speed, distance, and time, but also are widely used in the medical and engineering industry.

Let us learn more about rational functions along with how to graph them, their domain, range, asymptotes, etc along with solved examples.

What is a Rational Function?

A rational function is a function which is a ratio of polynomials. Any function of one variable, x, is called a rational function if it can be represented as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials such that q(x) ≠ 0. For instance, f(x) = (x² + x – 2) / (2x² – 2x – 3) is a rational function and here, 2x² – 2x – 3 ≠ 0.

We know that every constant is a polynomial and hence the numerators of a rational function can equally well represent constants. For instance, f(x) = 1/(3x+1) can be a rational function. But state that the denominators of rational functions cannot be constants. For instance, f(x) = (2x + 3) / 4 is NOT a rational function, rather, it is a linear function.

How to Identify a Rational Function?

By the definition of the rational function (from the previous section), if either the numerator or denominator is not a polynomial, then the fraction formed does NOT show a rational function. For instance, f(x) = (4 + √x)/(2-x), g(x) = (3 + (1/x)) / (2 – x), etc are NOT rational functions as numerators in these examples are NOT polynomials.

Domain and Range of Rational Function

Any fraction is not specified when its denominator is equal to 0. This is the key point that is used in seeing the domain and range of a rational function.

The domain of Rational Function

The domain of a rational function is the group of all x-values that the function can take. To see the domain of a rational function y = f(x):

• Select the denominator ≠ 0 and solve it for x.
• The set of all real numbers other than the values of x mentioned in the last step is the domain.

Instance: Find the domain of f(x) = (2x + 1) / (3x – 2).

Solution:

We selected the denominator not equal to zero.

3x – 2 ≠ 0
x ≠ 2/3

As a result,the domain = {x ∈ R | x ≠ 2/3}

Range of Rational Function

The range of a rational function is the group of all outputs (y-values) that it produces. To determine the range of a rational function y= f(x):

• If we have f(x) in the equation, substitute it with y.
• Solve the equation for x.
• Select the denominator of the resultant equation ≠ 0 and solve it for y.
• The set of all real numbers other than the values of y mentioned in the last step is the range.

Instance: Find the range of f(x) = (2x + 1) / (3x – 2).

Solution:

Let us replace f(x) with y. Then y = (2x + 1)/(3x – 2). Now, we will resolve this for x.

(3x – 2)y = (2x + 1)
3xy – 2y = 2x + 1
3xy – 2x = 2y + 1
x(3y – 2) = (2y + 1)
x = (2y + 1)/(3y – 2)

Now (3y – 2) ≠ 0
y ≠ 2/3

That’s why the range = {y ∈ R | y ≠ 2/3}

Graphing Rational Functions

Here are the phases for graphing a rational function:

• Recognize and draw the vertical asymptote using a dotted line.
• Identify and draw the horizontal asymptote with a dotted line.
• Plot the holes (if any)
• Find x-intercept (by using y = 0) and y-intercept (by using x = 0) and plot them.
• Draw a table of two columns x and y and establish the x-intercepts and vertical asymptotes in the table. Then bring some random numbers in the x-column on either side of each of the x-intercepts and vertical asymptotes.
• Calculate the corresponding y-values by substituting each of them in the function.
• You should plot all the points from the table and join them with curves without touching the asymptotes.

FAQs

Q: What is the Definition of a Rational Function?

A: A rational function is a function that is like a fraction where both the numerator and denominator are polynomials. It is like f(x) = p(x) / q(x), where both p(x) and q(x) are polynomials.

Q: What is the End Behaviour of Rational Function?

A: The end behaviour of parent rational function f(x) = 1/x:

• F(x) → 0 as x → ∞ or -∞ and this reaches the horizontal asymptote.
• F(x) → ∞ as x → 0⁺ and f(x) → -∞ as x → 0^– and these correspond to the vertical asymptote.

Q: How will you know if a function is rational?

A: Whenever a function has polynomials in its numerator and denominator, it is a rational function. But remember:

• The numerator of a rational function can be a constant value. For example, 1 / x² is a rational function.
• The denominator of a rational function cannot be a constant value. For example: x² / 1 is NOT a rational function.