SOLUTION: Find the inverse of the​ one-to-one function. f left parenthesis x right parenthesis equals 6 x plus 1 f Superscript negative 1 Baseline left parenthesis x right parenth

Algebra ->  Systems-of-equations -> SOLUTION: Find the inverse of the​ one-to-one function. f left parenthesis x right parenthesis equals 6 x plus 1 f Superscript negative 1 Baseline left parenthesis x right parenth      Log On


   

Question 1129946:
Find the inverse of the​ one-to-one function.
f left parenthesis x right parenthesis equals 6 x plus 1
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
nothing
​(Use integers or fractions for any numbers in the​ expression.)
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
-----
Find the inverse of the​ one-to-one function.
f left parenthesis x right parenthesis equals 6 x plus 1
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
nothing
-----

The given function should be represented f%28x%29=6x%2B1.

Finding Inverse of f:
Let f%5E-1%28x%29 be the inverse of f(x).
6%2A%28f%5E-1%28x%29%29%2B1=x
6%2A%28f%5E-1%28x%29%29=x-1
highlight%28f%5E-1%28x%29=%28x-1%29%2F6%29

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The usual way that students are taught to find the inverse of a function y=f(x) is to switch the x and y and solve for the new y. For this example it might look like this:

x+=+6y%2B1
x-1+=+6y
y+=+%28x-1%29%2F6

Tutor @josgarithmetic shows a different way of finding the inverse of a relatively simple function like this -- by solving the equation

(f(f^(-1)(x)) = x:

f%28f%5E%28-1%29%28x%29%29+=+x
6%28f%5E%28-1%29%28x%29%29%2B1+=+x
6%28f%5E%28-1%29%28x%29%29+=+x-1
f%5E%28-1%29%28x%29+=+%28x-1%29%2F6

That is a very good formal mathematical way to find the inverse of a simple function, based on the idea that f(f^(-1)(x)) = x.

If you don't need a formal derivation, you can do exactly the same thing as this second method without the formal mathematics, using the idea that the inverse function "gets you back where you started".

In this example, the given function takes an input value and does two things:
(1) multiply by 6; and
(2) add 1.

The inverse function, to get you back where you started, has to do the opposite operations in the opposite order:
(1) subtract 1; and
(2) divide by 6.

Using this informal method, we quickly see that the inverse function is

f%5E%28-1%29%28x%29+=+%28x-1%29%2F6

So now you have at your disposal three different ways to find the inverse of a relatively simple function:
(1) switch x and y and solve for the new y
(2) solve the equation f(f^(-1)(x)) = x for f^(-1)(x)
(3) do the opposite operations in the opposite order

Try using all three and find which one(s) work best for you....