Math CentralQuandaries & Queries


Question from san, a student:

what does inverse graph represent?

Hi San,

If you think of a function as a process where you give the function some input and it yields a result then the inverse of this function answers the question "Here is the result, what was the input?" For example if the function is f(x) = x2 then an inverse is g(x) = √x. If you give x = 5 to the function f it yields
f(5) = 52 = 25 and if you give x = 25 to the function g it yields g(25) = √25 = 5. (You may wonder why I didn't say ±5 but the symbol √ designates the positive square root.)

If you have studied logarithms then another example you have seen is f(x) = 10x and g(x) = log10(x).

In trigonometry an example is f(x) = tan(x) and g(x) = tan-1(x). If you have a right triangle and x is the measure of one of the other two angles then tan(x) is the ratio of the opposite side over the adjacent side. For the inverse function, if you have a right triangle and A is one of the vertices where the angle is not a right angle, let x be the ratio of the opposite side over the adjacent side. The measure of the angle at A is g(x) = tan-1(x).

One last example is the game show Jeopardy. In this game the contestants must perform an inverse operation. The host tells them an answer and they must give the question.


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