Question(4) $f: x \longmapsto 3+\frac{1}{x}$ find the rate for $f^{-1}$

Studdy Solution

STEP 1

What is this asking? We're looking for the rate of change of the inverse function of $f(x) = 3 + \frac{1}{x}$ .
Essentially, we need to find the derivative of the inverse function! Watch out! Don't mix up finding the inverse function with finding its derivative!
We need both here, but they're separate steps.
Also, remember that the rate of change of the inverse function isn't just the reciprocal of the original function's rate of change.

STEP 2

1. Find the inverse function
2. Calculate the derivative of the inverse

STEP 3

Let $y = f(x)$ .
So, we have $y = 3 + \frac{1}{x}$ .
Our goal is to solve for $x$ in terms of $y$ .

STEP 4

To isolate the term with $x$ , we subtract 3 from both sides: $y - 3 = \frac{1}{x}$ .

STEP 5

Now, we can take the reciprocal of both sides to get $x = \frac{1}{y - 3}$ .

STEP 6

So, the inverse function, $f^{-1}(y)$ , is given by $f^{-1}(y) = \frac{1}{y - 3}$ .
We can also write this as $f^{-1}(x) = \frac{1}{x - 3}$ by simply swapping the variable names.

STEP 7

Let's rewrite the inverse function as $f^{-1}(x) = (x - 3)^{-1}$ .
This makes it easier to apply the power rule for differentiation.

STEP 8

Using the power rule, we bring down the exponent and reduce it by one: $\frac{d}{dx} (x - 3)^{-1} = -1 \cdot (x - 3)^{-2}$ .

STEP 9

Don't forget the chain rule!
We need to multiply by the derivative of the inside function, which is $x - 3$ .
The derivative of $x - 3$ with respect to $x$ is just 1.
So, we multiply by 1: $-1 \cdot (x - 3)^{-2} \cdot 1$ .

STEP 10

This simplifies to $\frac{d}{dx} f^{-1}(x) = \frac{-1}{(x - 3)^2}$ .
This is the rate of change of our inverse function!

STEP 11

The rate of change of the inverse function $f^{-1}(x)$ is $\frac{-1}{(x - 3)^2}$ .

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