Definition
A rational function is essentially a function that can be expressed as the division of two polynomials. If you have two polynomials, P(x) and Q(x), a rational function R(x) can be represented as:
R(x) = P(x) / Q(x)
It's vital to remember that Q(x) should not be zero because in maths, dividing by zero is undefined. For a refresher on the fundamental concept of domain and range, see Domain and Range Basics.
Rational functions are sometimes termed as rational polynomial functions. However, it's crucial to distinguish between a "rational polynomial function" and a "polynomial with rational coefficients." The former pertains to our current discussion, while the latter refers to polynomials whose coefficients are rational numbers. For more details on polynomials, visit Quadratic Functions.
Vertical Asymptotes
- Understanding Vertical Asymptotes: These are vertical lines that the graph of a function approaches but never intersects. They represent values of x where the function becomes infinite.
- Finding Vertical Asymptotes: To pinpoint the vertical asymptotes of a rational function, set the denominator Q(x) to zero and solve for x. The solutions will indicate the x-values where the function is undefined, leading to vertical asymptotes.
- Example: Consider the function R(x) = x / (x2 - 1). To locate the vertical asymptotes, set x2 - 1 = 0. This results in x = 1 and x = -1 as potential vertical asymptotes.
Horizontal Asymptotes
- Understanding Horizontal Asymptotes: These are horizontal lines that the graph of a function approaches as x moves towards positive or negative infinity. Understanding how functions transform can provide deeper insights, as shown in Basic Transformations.
- Determining Horizontal Asymptotes: The method involves comparing the degrees of the polynomials in the numerator and the denominator:
- If the degree of P(x) is less than that of Q(x), then the x-axis (y = 0) is the horizontal asymptote.
- If both degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the two polynomials.
- If the degree of P(x) is greater than that of Q(x), the function does not have a horizontal asymptote.
- Example: For the function R(x) = (2x2 + 1) / (x2 + 3), since the degrees of the numerator and denominator are equal, the horizontal asymptote is y = 2.
Holes or Removable Discontinuities
- Understanding Holes: Holes, also termed as removable discontinuities, are specific points on the graph where the function is undefined. They arise due to factors that cancel out in both the numerator and the denominator.
- Identifying Holes: To detect holes in a rational function, factorise both the numerator and the denominator. If there are common factors between them, these factors cancel out, resulting in holes at the x-values where these factors equate to zero. Refer to Finding Local Extrema for further study.
- Example: For the function R(x) = x(x - 2) / (x - 2), the factor x - 2 cancels out from both the numerator and the denominator. This cancellation results in a hole at x = 2.
Practical Application
Rational functions aren't just theoretical constructs; they have practical applications in various domains, including physics, engineering, and economics. For instance, in electronics, rational functions can describe the relationship between voltage and current in certain circuits. In economics, they can model cost functions or supply and demand relationships under specific conditions. To explore other related concepts, visit the page on Basics of Rational Functions.
Example Question
Given the function R(x) = (x2 - 4) / (x - 2), determine its vertical and horizontal asymptotes and identify any holes.
Solution:
1. Vertical Asymptotes: Setting the denominator x - 2 to zero gives x = 2 as a potential vertical asymptote. However, since this factor also cancels out with a factor in the numerator, it results in a hole rather than an asymptote.
2. Horizontal Asymptotes: The degrees of the numerator and denominator are the same. Thus, the horizontal asymptote is the ratio of the leading coefficients, which is y = 1.
3. Holes: The factor x - 2 cancels out from both the numerator and the denominator, leading to a hole at x = 2.
FAQ
Rational functions are widely used in various real-world applications. For instance, in economics, they can model cost functions, revenue functions, or the relationship between supply and demand under certain conditions. In physics, they can represent relationships in circuits, particularly in the study of resonance in RLC circuits. In biology and medicine, they can model the rate of decay of substances or the growth and decay of populations. The versatility of rational functions makes them a valuable tool in representing scenarios where the relationship between variables can be expressed as a ratio of two polynomial functions.
Yes, a rational function can have more than one vertical asymptote. The number of vertical asymptotes is determined by the number of distinct real roots of the denominator that are not also roots of the numerator. For example, the rational function R(x) = 1 / (x2 - 4) has vertical asymptotes at x = 2 and x = -2, as these are the points where the denominator becomes zero. It's essential to remember that if a factor in the denominator also cancels out with a factor in the numerator, it results in a hole or removable discontinuity rather than a vertical asymptote.
No, not all rational functions are continuous everywhere. A rational function is discontinuous at any value of x where the denominator is zero and the numerator is non-zero. These points of discontinuity result in vertical asymptotes in the graph of the function. Additionally, if both the numerator and the denominator become zero at the same value of x, the function may have a removable discontinuity or a "hole" at that point. However, apart from these points of discontinuity, rational functions are continuous everywhere else in their domain.
The denominator of a rational function cannot be zero because division by zero is undefined in maths. When the denominator of a rational function is zero, the function itself becomes undefined at that particular value of x. This leads to vertical asymptotes in the graph of the function. In essence, as the value of the function approaches these points, the function's value tends to infinity or negative infinity, but it never actually reaches these points. This is why division by zero is prohibited, as it leads to undefined or infinite results.
A polynomial function is an algebraic expression that involves constants, variables, and non-negative integer exponents. It is a sum of terms, where each term is a monomial. For example, f(x) = 3x2 + 2x - 5 is a polynomial function. On the other hand, a rational function is a quotient of two polynomial functions. It is expressed as the division of one polynomial by another. For instance, R(x) = (x2 + 1) / (x - 3) is a rational function. While polynomial functions are defined for all real values of x, rational functions may have values of x for which they are undefined, typically where the denominator equals zero.
Practice Questions
The given function is R(x) = (x2 - 9) / (x - 3). To determine the vertical asymptotes, we need to set the denominator equal to zero. Solving x - 3 = 0, we get x = 3 as a potential vertical asymptote. However, factoring the numerator, we get x^2 - 9 = (x + 3)(x - 3). Since the factor x - 3 cancels out with the denominator, it results in a hole at x = 3 rather than an asymptote. The degrees of the numerator and denominator are the same, so the horizontal asymptote is the ratio of the leading coefficients, which is y = 1. In conclusion, the function has a horizontal asymptote at y = 1 and a hole at x = 3. There are no vertical asymptotes.
For the function R(x) = (2x2 + x - 3) / (x2 + 4), the degrees of the numerator and denominator are the same. This means the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 2 and that of the denominator is 1, so the horizontal asymptote is y = 2. As x approaches positive infinity, the function approaches the horizontal asymptote from above, and as x approaches negative infinity, the function approaches the horizontal asymptote from below. This means that the function will get closer and closer to the line y = 2 but will never touch or cross it as x becomes very large or very small.
Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.