A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator polynomial is not zero. In general, a rational function takes the form:
f(x) = P(x) / Q(x)
Here, P(x) and Q(x) are polynomials, and Q(x) ≠ 0. The numerator polynomial P(x) and the denominator polynomial Q(x) can have various degrees and coefficients.
Rational functions often have certain characteristics, such as vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts. These characteristics can be identified by analyzing the behavior of the function for different values of x.
Let's look at a couple of examples:
Example 1:
Consider the rational function f(x) = (3x^2 - 2x - 8) / (x - 2).
Here, the numerator polynomial is 3x^2 - 2x - 8, and the denominator polynomial is x - 2.
To find the vertical asymptote(s), we need to determine the value(s) of x for which the denominator Q(x) equals zero. In this case, we have x - 2 = 0, which gives x = 2. Therefore, the vertical asymptote is x = 2.
To find the x-intercept(s), we need to solve the equation P(x) = 0. In this case, we solve 3x^2 - 2x - 8 = 0. The solutions are x = -1 and x = 8/3. So, the x-intercepts are x = -1 and x = 8/3.
Example 2:
Consider the rational function g(x) = (2x + 1) / (x^2 + x - 6).
Here, the numerator polynomial is 2x + 1, and the denominator polynomial is x^2 + x - 6.
To find the vertical asymptote(s), we determine the values of x for which the denominator Q(x) equals zero. In this case, we need to solve x^2 + x - 6 = 0. Factoring the quadratic equation gives us (x - 2)(x + 3) = 0. So, the solutions are x = 2 and x = -3. Therefore, the vertical asymptotes are x = 2 and x = -3.
To find the x-intercept(s), we solve the equation P(x) = 0. In this case, we solve 2x + 1 = 0, which gives x = -1/2. So, the x-intercept is x = -1/2.
These are just a few examples illustrating the basic concepts of rational functions. Depending on the specific characteristics of the numerator and denominator polynomials, rational functions can exhibit a variety of behaviors. Analyzing their properties helps us understand their graphs and behavior in different regions of the coordinate plane.