Inverse Functions and Logarithms

Example (1.6.24) Find a formula for the inverse of the function

There are three steps to finding the inverse of a function:

Step 1)

Step 2) Solve for x in terms of y :

Step 3) Inter change x and y . This gives us

Let’s verify that we have the correct solution by checking the cancellation equations:

So the cancellation conditions are satisfied, and we have found the inverse function correctly.

Example Find a formula for the inverse of the function f (x) = 5 − 4x³.

There are three steps to finding the inverse of a function:

Step 1) f (x) = y = 5 − 4x³.

Step 2) Solve for x in terms of y:

Step 3) Interchange x and y. This gives us

Let’s verify that we have the correct solution by checking the cancellation equations:

So the cancellation conditions are satisfied, and we have found the inverse function correctly.

Example (1.6.28) Find a formula for the inverse of the function

There are three steps to finding the inverse of a function:

Step 1) .This was already d one for us .

Step 2) Solve for x in terms of y:

Step 3) Interchange x and y. This gives us

Let’s verify that we have the correct solution by checking the cancellation equations:

So the cancellation conditions are satisfied, and we have found the inverse function correctly.

Example (1.6.43) Graph the given functions on a common screen . How are these graphs related?



The functions can be graphed using the following Mathematica commands :

Plot[{ Log [1.5, x], Log [x], Log[10, x], Log[50, x]}, {x, -5, 5},
PlotRange -> {{-1, 3}, {-3, 4}}]

I added options to get the plots to be different colours, and to have the axes labeled. The commands I used
to generate the plot be low was :

Plot[{Log[1.5, x], Log [x], Log[10, x], Log[50, x]}, {x, -5, 5},
PlotRange -> {{-1, 3}, {-3, 4}}, AxesLabel -> {"x", "Exp [x]"},
PlotStyle ->
{{RGBColor[1, 0, 0]}, {RGBColor[0, 1, 0]},
{RGBColor[0, 0, 1]}, {RGBColor[1, 1, 0]}}]

In my plots, the functions are:
 

	red
	green
	blue
	yellow

All the plots pass through the point (1, 0), all increase, and all approach negative infinity as x approaches
zero from the left . As the base increases, the function stays closer to zero.

Example (1.6.61) Starting with the graph of y = ln x, find the equation of the graph that results from

a) shifting 3 units upward,
b) shifting 3 units to the left,
c) reflecting about the x-axis,
d) reflecting about the y-axis,
e) reflecting about the line y = x,
f) reflecting about the x-axis and then about the line y = x,
g) reflecting about the y-axis and then about the line y = x.

I have plotted the graphs with the reflections below.