Rational Function in Maths

A rational function is a polynomial ratio in which the denominator polynomial is not equal to zero. Isn’t it similar to the definition of a rational number (of the type p/q, where q is not equal to 0)? Did you know that rational functions are used in a variety of sectors in our daily lives? They are commonly used in the medical and technical industries to define the connection between speed, distance, and time.

Definition of a Rational Function

A rational function is one in which the ratio of polynomials is the same. A rational function is one with only one variable, x, and may be written as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials with q(x) 0. For instance, f(x) = (x2 + x – 2) / (2x2 – 2x – 3) is a rational function, and 2x2 – 2x – 3 is equal to 0.

Because every constant is a polynomial, the numerators of a rational function can also be constants. F(x) = 1/(3x+1) is an example of a rational function. The denominators of rational functions, however, cannot be constants. For instance, f(x) = (2x + 3) / 4 is a linear function, not a rational function.

What Makes a Rational Function?

If either the numerator or denominator is not a polynomial, the fraction created does not reflect a rational function, according to the preceding section’s definition. f(x) = (4 + x)/(2-x), g(x) = (3 + (1/x)) / (2 – x), and so on are not rational functions since the numerators are not polynomials.

Rational Function Domain and Range

When the denominator of a fraction equals 0, it is not defined. This is the crucial element in determining a rational function’s domain and range.

• Rational Function Domain

The collection of all x-values that a rational function may take is known as its domain. To determine the domain of a rational function y = f(x), use the following formula:

Set the denominator not equal to zero and solve the equation to get the value for x.

The domain is the set of all real numbers other than the x values indicated in the previous step.

• Rational Function Range

A rational function’s range is the collection of all its outputs (y-values). To determine the range of a rational function y= f(x), use the following formula:

Replace f(x) with y if it appears in the equation.

For x, solve the equation.

Set the resultant equation’s denominator not equal to 0 and solve it for y.

The range is the collection of all real numbers other than the y values given in the previous phase.

Rational Function Asymptotes

There are three sorts of asymptotes for a rational function: horizontal, vertical, and slant asymptotes. It may also contain holes in addition to this. Let’s look at how to locate each of them.

A Rational Function’s Holes

A rational function’s holes are points that appear to be present on the graph of the rational function but are not. They may be found by solving for x and setting the linear factors that are common factors of both the numerator and denominator of the function to zero. By inserting the x-values in the simplified function, we can obtain the appropriate y-coordinates of the spots. There are no holes required in any rational function. Only when the numerator and denominator have linear common factors can holes appear.

A Rational Function’s Vertical Asymptote

A function’s vertical asymptote (VA) is an imaginary vertical line that its graph approaches but never touches. It has the formula x = a number. “Some number” is directly associated with the domain’s excluded values. However, if there is a hole at x = some integer, there cannot be a vertical asymptote at that number. One or more vertical asymptotes may exist for a rational function. To obtain the vertical asymptotes of a rational function, follow these steps:

To cancel all common components, simplify the function first (if any).

Set the numerator to 0 and find (x) (or equivalently just get the excluded values from the domain by avoiding the holes).

A Rational Function’s Horizontal Asymptote

An imaginary horizontal line to which a function’s graph looks to be extremely close but never touches is called a horizontal asymptote (HA). It has the formula y = a number. “Some number” is directly associated with the range’s excluded values. There can only be one possibility of horizontal asymptote for a rational function. Using the degrees of the numerator (N) and denominators (D) is a simple approach to locate the horizontal asymptote of a rational function (D).

• If N D, then at y = 0 there is a HA.

• If N exceeds D, there is no HA.

• The HA is the y = ratio of the leading coefficients if N = D.

Rational Function Slant (Oblique) Asymptotes

A slant asymptote is an imaginary oblique line that appears to touch a portion of the graph. Only when the degree of the numerator (N) is exactly one more than the degree of the denominator can a rational function have a slant asymptote (D). Its equation is y = quotient, which is produced by using long division to divide the numerator by the denominator.

Inverse of Rational Function

To get the inverse of a rational function y = f(x), use the following formula:

Substitute y for f(x).

Swap the x and y coordinates.

For y, solve the resultant equation.

The inverse f-1(x) would be the outcome.

Conclusion

The form of a rational function equation is f(x) = P(x) / Q(x), where Q(x) is not equal to 0.At least one vertical asymptote exists for every rational function. There is only one horizontal asymptote for any rational function. There is only one slant asymptote for any rational function. The excluded values of a rational function’s domain aid in the identification. The excluded values of a rational function’s range aid in the identification of Has. The holes would be created by the linear elements that are cancelled when a rational function is simplified.