When a function is defined in terms of a rational expression in one variable, it is known as a rational
function. The value of the denominator in the expression cannot be zero. Some examples of rational functions are as follows:
f(x) = (5x + 6)/2x, g(x) = (0.45 x – 7)/x, h(x)= (9 – 13x)/(3x +5)
Suppose the cost of renting a car is $50 for the first day and $25 for each additional day. The average cost per day for renting the car is given by the equation C = (50 + 25d)/(d + 1), where d is the number of days after the first day. The function that gives the average daily cost can be written as C(d) = (50 + 25d)/(d + 1).
Example: Using the function C(d) above, find the average cost when renting a car for 4 days and for renting a car for 7 days. To find the average cost of the rental per day for 4 days, substitute 3 for d and solve.
C(3) = (50 + 25(3))/(3 + 1) = (50 + 75)/4 = 125/4 = $31.25.
The average cost per day for a 7 day rental is C(6) = (50 + 25(6))/(6 + 1) = $28.57.
To graph the function, set up a table and choose values for d and solve for C(d) and plot the points on the graph. When doing so, you will notice how C(d) approaches 25 as d gets larger. Also, it will become evident how C(d) approaches 50 as d gets smaller. When a graph approaches a line, the line it approaches is called an asymptote. The line d = 25 is called the horizontal asymptote. If C(d) = y, then the line y = 50 is called the vertical asymptote.
Example: The time (in hours) it takes for an airplane to travel 3,000 miles is given by the function r(t) = 3000/t. We can solve for the time t when given the following rates r.
If r(t) = 550, then t is found by taking 550 = 3000/t. Therefore, t = 5.45 hours. If f(t) = 475, then t is found by taking 475 = 3000/t. Therefore, t = 6.32 hours. If r(t) = 625, then t is found by taking 625 = 3000/t. Therefore, t = 4.8 hours.
Suppose you have the graph of f(x) = 2/x. From the graph you can find approximate value for f(1), f(2) and so on. To find f(1) move over 1 on the x axis and up on the f(x) axis until you hit the graph. Suppose the value of f(1) = 2. Likewise, for f(2), move over 2 on the x axis from the origin and move up until you hit the graph. Suppose the value of f(2) = 1 and similarly the value of f(4) = ½. The horizontal asymptote is the line x = 0 and the vertical asymptote is the line y = 0.
Many times it’s important to find the domain of a function. Recall that the denominator of a fraction or rational expression is undefined when it equals zero. To find the domain of a rational function, set the denominator equal to zero and solve for the variable. The value or values you obtain are excluded from the domain and all other values are included in the domain.
Example: Find the domain of f(x) = 5/(x + 3).
Set the denominator equal to zero and solve for x. Therefore, x + 3= 0, and x = -3. The domain is all real numbers except for -3. In interval notation, the domain is (-∞, -3) U (-3, ∞).
Example: Find the domain of f(x) = (3x – 7)/(x2 – 11x + 18).
Set the denominator equal to zero and solve for x. Therefore, x2 – 11x + 18 = 0. Factor to get (x -2)(x – 9) = 0. Solving gives you x = 2, x =9. That means that 2 and 9 are not part of the domain. The domain is all real numbers except for 2 and 9.
This guide should help assist students having difficulty understanding the concept of rational functions.