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