Rational functions

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form of a rational function is:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials. The domain of a rational function is all real numbers except where the denominator Q(x) is equal to zero.

Identifying Rational Functions

To identify a rational function, check if it can be written in the form of a fraction where both the numerator and the denominator are polynomials. For example:

• f(x) = (2x^2 + 3x + 1) / (x - 2) is a rational function.
• g(x) = x^2 + 1 is not a rational function because it is not in fraction form.

Finding the Domain

The domain of a rational function includes all real numbers except where the denominator equals zero. To find the domain, follow these steps:

1. Set the denominator Q(x) equal to zero.
2. Solve for x to find the values that are excluded from the domain.

For example, consider the function:

f(x) = (3x + 4) / (x^2 - 1)

To find the domain:

1. Set the denominator equal to zero: x^2 - 1 = 0.
2. Factor the equation: (x - 1)(x + 1) = 0.
3. Set each factor to zero: x - 1 = 0 or x + 1 = 0.
4. Thus, x = 1 or x = -1.

The domain of f(x) is all real numbers except x = 1 and x = -1. In interval notation, the domain is:

(−∞, -1) ∪ (-1, 1) ∪ (1, ∞)

Vertical Asymptotes

A vertical asymptote is a line x = a where the function approaches infinity or negative infinity. Vertical asymptotes occur at the values of x that make the denominator zero (and are not cancelled by the numerator). To find vertical asymptotes:

1. Set the denominator Q(x) equal to zero.
2. Solve for x.

Using the previous example f(x) = (3x + 4) / (x^2 - 1), we found that the vertical asymptotes are at x = 1 and x = -1.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line y = b that the graph approaches as x approaches infinity or negative infinity. To find horizontal asymptotes, compare the degrees of the polynomials in the numerator and the denominator:

• If the degree of P(x) < degree of Q(x), then y = 0 is the horizontal asymptote.
• If the degree of P(x) = degree of Q(x), then y = a/b, where a and b are the leading coefficients of P(x) and Q(x), respectively.
• If the degree of P(x) > degree of Q(x), then there is no horizontal asymptote.

For the function f(x) = (3x + 4) / (x^2 - 1):

• The degree of the numerator (3x + 4) is 1.
• The degree of the denominator (x^2 - 1) is 2.

Since the degree of P(x) < degree of Q(x), the horizontal asymptote is:

y = 0.

Graphing Rational Functions

To graph a rational function, follow these steps:

1. Determine the domain.
2. Find the vertical and horizontal asymptotes.
3. Identify any x-intercepts and y-intercepts.
4. Plot the asymptotes and intercepts on the graph.
5. Sketch the graph considering the asymptotes and intercepts.

For example, let us graph the function f(x) = (3x + 4) / (x^2 - 1).

Step 1: Determine the Domain

The domain is all real numbers except x = 1 and x = -1.

Step 2: Find Vertical Asymptotes

The vertical asymptotes are x = 1 and x = -1.

Step 3: Find Horizontal Asymptote

The horizontal asymptote is y = 0.

Step 4: Identify Intercepts

To find the x-intercept, set the numerator equal to zero:

3x + 4 = 0 → x = -4/3.

The x-intercept is (-4/3, 0).

To find the y-intercept, set x = 0:

f(0) = (3(0) + 4) / (0^2 - 1) = 4 / -1 = -4.

The y-intercept is (0, -4).

Step 5: Sketch the Graph

Now, plot the vertical asymptotes at x = 1 and x = -1, the horizontal asymptote at y = 0, and the intercepts (-4/3, 0) and (0, -4). The graph will approach the asymptotes but never touch them.

Check Your Understanding

• What is the domain of the function f(x) = (2x + 1) / (x^2 - 4)?
• Find the vertical asymptotes of the function g(x) = (x^2 + 3) / (x^2 - 9).
• Determine the horizontal asymptote of the function h(x) = (4x^3 - 2) / (2x^3 + 5).
• Sketch the graph of the function k(x) = (x - 1) / (x^2 + 2x - 3).