BrainDen.com - Brain Teasers
• 0

# Ridiculous Inverse

## Question

Many of my algebra and precalculus students think the 'inverse function' of f(x), often written f^(-1)(x), is the same as the reciprocal 1/f(x) (mistaking the -1 for an exponent). This (as I am obliged to remind them) is almost always false. But can you find at least one function whose inverse is also its reciprocal? Tiebreaker: Find as many as you can!
• 1
• 1

## Recommended Posts

• 0

For starters,

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

but you told us that:
f^-1(f(x)) = 1/f(x)

putting these together,
f(x) = 1/x

Other than tricky ones, to me seems to be the only such function... Thinking of tricky ones now ...
• 1
• 1
##### Share on other sites

• 0

There are a few functions that are their own inverses. 1/x (away from 0) is one instance.

For tiebreaker are you asking for other instances of functions for which f[f(x)]= x?

Or for more functions whose inverses are their reciprocals?

##### Share on other sites

• 0

There are a few functions that are their own inverses. 1/x (away from 0) is one instance.

For tiebreaker are you asking for other instances of functions for which f[f(x)]= x?

Or for more functions whose inverses are their reciprocals?

The latter

##### Share on other sites

• 0

this obviously only applies to invertible functions and since invertible functions have exactly one inverse, I claim 1/x is the only such function.

For starters,

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

but you told us that:

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

putting these together,

f(x) = 1/x

Other than tricky ones, to me seems to be the only such function... Thinking of tricky ones now ...

##### Share on other sites

• 0

this obviously only applies to invertible functions and since invertible functions have exactly one inverse, I claim 1/x is the only such function.

For starters,

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

but you told us that:

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

putting these together,

f(x) = 1/x

Other than tricky ones, to me seems to be the only such function... Thinking of tricky ones now ...

I think the OP means that f^-1(x)=1/f(x).

One group fo solutions would be ones of the form (ax+b)/(cx+d) for appropriate a,b,c,d.

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.