In Exercises 35-38 : a. Find f^-1(x) b. Graph f and f^-1...

In Exercises 35-38 :
a. Find $f^{-1}(x)$
b. Graph $f$ and $f^{-1}$ together.
c. Evaluate $d f / d x$ at $x=a$ and $d f^{-1} / d x$ at $x=f(a)$ to show that at these points $d f^{-1} / d x=1 /(d f / d x)$
$$f(x)=(1 / 5) x+7, \quad a=-1$$

Key Concepts

Evaluating Derivatives at Specific Points

When evaluating the derivative of a function or its inverse at specific points, it is important to carefully compute f'(a) for the original function and then use the inverse derivative formula (f?¹)'(f(a)) = 1 / f'(a). This step underlines the interplay between a function's local behavior and that of its inverse.

Derivative of the Inverse Function

The derivative of an inverse function at a point is given by the reciprocal of the derivative of the original function at the corresponding point. More precisely, if f is differentiable and invertible, then (f?¹)'(x) = 1 / f'(f?¹(x)). This relationship is a direct consequence of the inverse function theorem.

Finding the Inverse of a Function

Finding an inverse involves switching the roles of the dependent and independent variables and then solving for the new dependent variable. This process confirms that the function is invertible by ensuring that each output is uniquely produced by one input.

Graphical Relationship Between a Function and Its Inverse

The graph of a function and its inverse are symmetric with respect to the line y = x. By reflecting the graph of the function across this line, one can visualize the graph of its inverse.

Inverse Functions

An inverse function essentially reverses the mapping of the original function, meaning if f maps an input x to an output y, then the inverse function, f?¹, maps y back to x. In other words, f(f?¹(x)) = x and f?¹(f(x)) = x, provided that f is one-to-one and onto in the domain of interest.

Transcript

00:01 So our current question is we're going f of x equals 1 over 5x plus 7, and we're asked to find the inverse, then graph both equations, then show the derivative of the inverse is the same as 1 over the derivative of the original function at x equals negative 1 for the original function.

00:22 So to start with finding the inverse should be pretty simple.

00:27 Just swap your x and y and solve for y.

00:29 We have x equals 1 over 5, y plus 7, then subtract both sides by 7, and then multiply both sides by 5.

00:44 Don't forget we're multiplying the entire sides of both sides, which means that we will be distributing.

00:52 So we end up with y equals 5x minus 35.

01:01 And that should be our inverse function.

01:07 For the second part, now we just graph up these functions.

01:14 So if i'm trying to graph, say, the second one, our awareness up to set negative 35, and the slope is 5.

01:21 So each time i go right one, i go up 5.

01:24 Right 1, 5.

01:32 And if i go left one, i would go down 5.

01:36 So this graph, this line should look something like this, and then for our first function, our why under subset 7, and now each time we go right one, we go down one, and we go up 1 5th.

01:55 But it might be easier instead to go right 5, so then we just go up 1.

02:01 If you think about rise over run, this is exactly what's happening here.

02:06 So we go left, left 5, i go down 1.

02:23 So these should be both of our functions...

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