Question Video: Determining Which of a Group Functions Is Not Its Own Inverse Mathematics

Which of the following functions is NOT its own inverse? [A] 𝑓(π‘₯) = βˆ’8 βˆ’ π‘₯ [B] 𝑓(π‘₯) = βˆ’(8/π‘₯) [C] 𝑓(π‘₯) = 8π‘₯ [D] 𝑓(π‘₯) = π‘₯ [E] 𝑓(π‘₯) = βˆ’(4/π‘₯).

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Video Transcript

Which of the following functions is not its own inverse? Is it A) 𝑓 of π‘₯ equals negative eight minus π‘₯, B) 𝑓 of π‘₯ equals negative eight over π‘₯, C) 𝑓 of π‘₯ equals eight π‘₯, D) 𝑓 of π‘₯ equals π‘₯, or E) 𝑓 of π‘₯ equals negative four over π‘₯?

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 π‘₯ equals negative eight minus π‘₯. What is the output of this function when the input is one? Replacing π‘₯ 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 π‘₯, the output is negative eight minus π‘₯. 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 π‘₯ is negative eight minus negative eight minus π‘₯. Here, we’ve just replaced the π‘₯ in the definition of 𝑓 of π‘₯ with negative eight minus π‘₯. Expanding the bracket, we get negative eight plus eight plus π‘₯, which is π‘₯. So we do get our input back. For any input π‘₯, the output is negative eight minus π‘₯ 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 π‘₯ equals negative eight over π‘₯. For an input of π‘₯, we get an output of negative eight over π‘₯. Feeding this output back into the function, we get 𝑓 of negative eight over π‘₯. And replacing π‘₯ by negative eight over π‘₯ in the definition of 𝑓 of π‘₯, we get 𝑓 of negative eight over π‘₯ equals negative eight over negative eight over π‘₯. Multiplying both numerator and denominator by π‘₯, we get negative eight π‘₯ over negative eight, which simplifies to just π‘₯. 𝑓 takes an input of π‘₯ to an output of negative eight over π‘₯. 𝑓 also takes the output value negative eight over π‘₯ back to the input value π‘₯. So just like in option A, we can see that the function 𝑓 of π‘₯ in option B is its own inverse.

Just a quick note to say that we didn’t have to worry about π‘₯ over π‘₯ being zero over zero because π‘₯ equals zero is not in the domain of our function.

We move on to option C) 𝑓 of π‘₯ equals eight π‘₯. So an input of π‘₯ is taken to an output of eight π‘₯. Where does 𝑓 take this output, eight π‘₯? It takes it to eight times eight π‘₯, which is 64π‘₯. This is not the input value π‘₯ that we started with. For a concrete example, let’s take π‘₯ 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 π‘₯ equals eight π‘₯ is not its own inverse.

With a bit of work, you can see that for option C, 𝑓 inverse of π‘₯ is equal to π‘₯ over eight. We should just quickly check that the functions in options D and E are their own inverse. An input of π‘₯ gives an output of π‘₯. And when feeding that output π‘₯ 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 π‘₯ equals eight π‘₯.

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