SOLUTION: Find the inverse of the function from the ‘procedural perspective’ discussed in Example 6.1.5 {{{matrix(1,3, "f(x)", ""="",log(2,(10x-20)))}}}

Algebra ->  Inverses -> SOLUTION: Find the inverse of the function from the ‘procedural perspective’ discussed in Example 6.1.5 {{{matrix(1,3, "f(x)", ""="",log(2,(10x-20)))}}}       Log On


   

Question 1125603: Find the inverse of the function from the ‘procedural perspective’ discussed in Example 6.1.5

matrix%281%2C3%2C%0D%0A%22f%28x%29%22%2C+%22%22=%22%22%2Clog%282%2C%2810x-20%29%29%29
Found 3 solutions by Edwin McCravy, MathLover1, greenestamps:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
matrix%281%2C3%2C%0D%0A%22f%28x%29%22%2C+%22%22=%22%22%2Clog%282%2C%2810x-20%29%29%29

That has the red graph, with the blue vertical asymptote:




Replace f(x) by y

matrix%281%2C3%2C%0D%0A%22y%22%2C+%22%22=%22%22%2Clog%282%2C%2810x-20%29%29%29

Interchange x and y

matrix%281%2C3%2C%0D%0A%22x%22%2C+%22%22=%22%22%2Clog%282%2C%2810y-20%29%29%29

Solve for y.

Raise 2 to the both sides power:

matrix%281%2C3%2C%0D%0A2%5Ex%2C+%22%22=%22%22%2C2%5E%28log%282%2C%2810y-20%29%29%29%29

Use a rule of logs a%5E%28+log%28a%2C%28b%29%29%29+=b on the right side:

matrix%281%2C3%2C%0D%0A2%5Ex%2C+%22%22=%22%22%2C10y-20%29

matrix%281%2C3%2C%0D%0A-10y%2C%22%22=%22%22%2C-2%5Ex-20%29

Divide both sides by -10



matrix%281%2C3%2C%0D%0Ay%2C%22%22=%22%22%2C2%5Ex%2F10%2B2%29

Replace y by f-1(x)

f-1 =2%5Ex%2F10%5E%22%22%2B2   <--- the answer!

This has the green graph with the blue horizontal asymptote:



Put the original red function together with its green inverse on the 
same set of axes:



Now if we draw the identity line (dotted) whose equation is y = x,


 
we see that the green inverse function is the red original function
reflected across the dotted identity line which has equation y = x.

Edwin

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

+f%28x%29=log%282+%2C%2810x-20%29%29+
change the base, replace f%28x%29+with y:

+y=log%28%2810x-20%29%29%2F+log%28%282%29%29
Interchange the variables:
+x=log%28%2810y-20%29%29%2F+log%28%282%29%29
solve or y
+x%2Alog%28%282%29%29=log%28%2810y-20%29%29
+log%28%282%5Ex%29%29=log%28%2810y-20%29%29 ...since log equals, we will have
+2%5Ex=10y-20
.+2%5Ex%2B20=10y
+y=%282%5Ex%2B20%29%2F10
+y=%281%2F10%29%282%5Ex%2B20%29->your inverse
so, replace y with +f%5E-1%28x%29:
+f%5E-1%28x%29=%281%2F10+%29%282%5Ex+%2B+20%29


Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


I don't know what "the ‘procedural perspective’ discussed in Example 6.1.5" is....

Two other tutors have given you good responses showing how you can find the inverse of a function by switching the x and y and solving for the new y. But that might not be what you were looking for.

The "procedural perspective" suggests to me that you might have been looking for something like the following.

An inverse function "gets you back where you started." For relatively simple functions, that means performing the opposite operations and in the opposite order.

So look at what the given function does with its input. (To help with this part of the analysis, think of the order of operations you would do to evaluate the function for a given input.)

(1) multiply by 10; then
(2) subtract 20; then
(3) take logarithm base 2

The inverse function does the opposite operations, in the opposite order:

(1) raise 2 to the given power (the function at this point is 2%5Ex); then
(2) add 20 (the function is now 2%5Ex%2B20; and last
(3) divide by 10

The inverse function is

%282%5Ex%2B20%29%2F10