Video Transcript
Which of the following functions is
not its own inverse? Is it A) 𝑓 of 𝑥 equals negative
eight minus 𝑥, B) 𝑓 of 𝑥 equals negative eight over 𝑥, C) 𝑓 of 𝑥 equals eight
𝑥, D) 𝑓 of 𝑥 equals 𝑥, or E) 𝑓 of 𝑥 equals negative four over 𝑥?
A function 𝑓 takes an input value
to an output value. The inverse function of 𝑓 reverses
this process taking an output value back to its input. This question is about functions
which are their own inverse. That is, functions 𝑓 which the
inverse function, written 𝑓 superscript negative one which takes the output value
back to the input, is just the same as function 𝑓. So we can get rid of this
superscript negative one.
Let’s try the function in option A)
𝑓 of 𝑥 equals negative eight minus 𝑥. What is the output of this function
when the input is one? Replacing 𝑥 by one, we get
negative eight minus one, which is negative nine. So the function 𝑓 takes the input
one to negative nine. Does the function 𝑓 take the
output negative nine back to the input one? 𝑓 needs to do this if it is its
own inverse. Let’s try it out. 𝑓 of negative nine is equal to
negative eight minus negative nine, which is one. So 𝑓 does take the output negative
nine back to one. But it’s not enough for 𝑓 to take
one output back to its input; it has to do this for all pairs of inputs and
outputs.
For a general input 𝑥, the output
is negative eight minus 𝑥. Just like with the numerical case,
we feed this output back into 𝑓 and see if we get our input back. 𝑓 of the output negative eight
minus 𝑥 is negative eight minus negative eight minus 𝑥. Here, we’ve just replaced the 𝑥 in
the definition of 𝑓 of 𝑥 with negative eight minus 𝑥. Expanding the bracket, we get
negative eight plus eight plus 𝑥, which is 𝑥. So we do get our input back. For any input 𝑥, the output is
negative eight minus 𝑥 and feeding that output back into 𝑓 gives us our input
back. So 𝑓 is its own inverse. This is therefore not the function
we’re looking for; we’re looking for the function which is not its own inverse.
We move on to try option B) 𝑓 of
𝑥 equals negative eight over 𝑥. For an input of 𝑥, we get an
output of negative eight over 𝑥. Feeding this output back into the
function, we get 𝑓 of negative eight over 𝑥. And replacing 𝑥 by negative eight
over 𝑥 in the definition of 𝑓 of 𝑥, we get 𝑓 of negative eight over 𝑥 equals
negative eight over negative eight over 𝑥. Multiplying both numerator and
denominator by 𝑥, we get negative eight 𝑥 over negative eight, which simplifies to
just 𝑥. 𝑓 takes an input of 𝑥 to an
output of negative eight over 𝑥. 𝑓 also takes the output value
negative eight over 𝑥 back to the input value 𝑥. So just like in option A, we can
see that the function 𝑓 of 𝑥 in option B is its own inverse.
Just a quick note to say that we
didn’t have to worry about 𝑥 over 𝑥 being zero over zero because 𝑥 equals zero is
not in the domain of our function.
We move on to option C) 𝑓 of 𝑥
equals eight 𝑥. So an input of 𝑥 is taken to an
output of eight 𝑥. Where does 𝑓 take this output,
eight 𝑥? It takes it to eight times eight
𝑥, which is 64𝑥. This is not the input value 𝑥 that
we started with. For a concrete example, let’s take
𝑥 equals one. Giving an input of one to the
function 𝑓 produces an output of eight, and the inverse function therefore would
take an input eight and return at the output one. The inverse function will take the
output eight back to the input one. But we can see that 𝑓 of eight is
eight times eight, which is 64. 𝑓 inverse is therefore not the
same function as 𝑓. For this function, 𝑓 is not its
own inverse. This is therefore the answer to our
question. The function 𝑓 of 𝑥 equals eight
𝑥 is not its own inverse.
With a bit of work, you can see
that for option C, 𝑓 inverse of 𝑥 is equal to 𝑥 over eight. We should just quickly check that
the functions in options D and E are their own inverse. An input of 𝑥 gives an output of
𝑥. And when feeding that output 𝑥
into the function 𝑓, we get that input back. So the function in option D is its
own inverse. And the process for showing that
the function in option E is its own inverse is exactly the same as it was for option
B, except with four instead of eight.
We therefore conclude that of our
five options, the only function which is not its own inverse is the function in
option C) 𝑓 of 𝑥 equals eight 𝑥.
