Example 8 from Algebra and Trigonometry, 5.7 Inverses and Radical Functions

Finding the Inverse of a Rational Function

The function $C = \frac{20 + 0.4 n}{100 + n}$ represents the concentration $C$ of an acid solution after $n$ mL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for $n$ in terms of $C .$ Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We first want the inverse of the function in order to determine how many mL we need for a given concentration. We will solve for $n$ in terms of $C .$

$\begin{matrix} C & = & \frac{20 + 0.4 n}{100 + n} \\ C (100 + n) & = & 20 + 0.4 n \\ 100 C + C n & = & 20 + 0.4 n \\ 100 C - 20 & = & 0.4 n - C n \\ 100 C - 20 & = & (0.4 n - C) n \\ n & = & \frac{100 C - 20}{0.4 - C} \end{matrix}$

Now evaluate this function at 35%, which is $C = 0.35.$

$\begin{matrix} n & = & \frac{100 (0.35) - 20}{0.4 - 0.35} \\ = & \frac{15}{0.05} \\ = & 300 \end{matrix}$

We can conclude that 300 mL of the 40% solution should be added.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the inverse of the function $f (x) = \frac{x + 3}{x - 2} .$

How did it go?

Solution

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Solution

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