Finding the Inverse of a Nonlinear Rational Function
Review Problem: Let
- State the domain and range of R.
- Is R a function?
- Is R a one-to-one function?
- Find , the inverse of R.
- State the domain and range of .
So how do we find the inverse of a rational function? We can apply what we have already learned about functions and their inverses! Remember, once we have confirmed that our function is one-to-one, we can use the following steps to find its inverse:
- Change to y if necessary.
- Interchange x and y.
- Solve for y in terms of x.
- Change y to.
Example: Let .
- State the domain of .
- Verify that is a one-to-one function.
- Find .
- State the domain and range of .
Application of the Inverse of a Rational Function:
Since most pharmaceutical reference manuals only list adult dosages, pediatricians have to be especially careful when calculating dosages for their patients. Fortunately, there are several methods to choose from when calculating how much of a particular antibiotic or medication should be prescribed to a child. In this activity, we focus on a method known as Young’s Rule.
According to Young’s Rule, a child’s dosage (in mg) can be calculated using the rational function, where D is the adult dosage (in mg), and x is the child’s age (in years).
- Suppose the adult dosage of an antibiotic is 100 mg per day. Use Young’s Rule to determine the corresponding children’s dosage for the given ages.
Child’s Age, x / Dosage,
2
6
12
- Find .
- What does each variable of the inverse function represent?
- Use the inverse function to determine the age of a child who has been prescribed 40 mg of an antibiotic.