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If two functions, f(x) and g(x), are inverse functions, what must the composition of f(g(x)) and g(f(x)) both equal?

A. 0
B. 1
C. f(x) = g(x)
D. x

1 Answer

6 votes

Answer: Choice D) x

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Step-by-step explanation:

The composition of the original function with its inverse will cancel out the operations to yield the input as the output.

Consider the example functions shown below

  • f(x) = x+5
  • g(x) = x-5

The first function adds 5 while the second function does the complete opposite and subtract 5. The two functions are inverses of each other.

We see that,

f(x) = x+5

f(g(x)) = g(x)+5

f(g(x)) = x-5+5

f(g(x)) = x

and you should find that g(f(x)) = x for similar steps.

No matter what the x input is, the output will be identical to the input.

For instance, if we plug in x = 7, then adding 5 to it gets us 12. Then undoing that operation to subtract 5 gets us back to 7 again. This is one example showing why both f(g(x)) and g(f(x)) both equal x, where f and g are inverses of each other.

User Jim Meyer
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