Some function f(x) and its derivative f '(x) are continuous and differentiable for all real numbers, and some of the values for the functions are given in the below table:

Based on the information given, answer the following questions:

(a) Evaluate .

(b) At what value c is the graph of discontinuous?

You may not use a graphing calculator • Difficulty:

Solution

(a) Remember that , allowing us to find the derivative of an inverse function given only the original function. However, you'll need to be able to compute . Remember, a function and its inverse differ in that the input for one is the output for the other. Since the above table tells us that f(1) = 3, we can be sure that . Plug what you know into the formula we began part (a) with and we get

The table tells us that f '(1) = –2, so .

(b) The formula we used in part (a), written more generally, says that . Therefore, whenever the denominator equals 0, the fraction will be undefined. So, the answer to our question is the equivalent to the solution of the equation . To solve this, first decide at what value of x does f ' equal 0. The only answer we can be sure of is when x = 2. Since f '(2) = 0, we can then say, by substitution, that . This means the exact same thing as f(2) = a. Basically, when it's all said and done, we're just looking for f(2), so the answer is –1. To check, try and evaluate :

Obviously, that derivative doesn't exist.