Question

If a function f has an inverse, then f⁻¹(f(x)) = x?
1) True
2) False

Answer 1

It is true that if a function f has an inverse, then f⁻¹(f(x)) = x. This is the fundamental characteristic of inverse functions, which applies to various mathematical operations and exemplifies how inverse operations 'undo' each other.

Step-by-step explanation:

The question ponders whether it is true or false that if a function f has an inverse, then f⁻¹(f(x)) = x. It is indeed true that if a function f has an inverse f⁻¹, then applying f⁻¹ to the result of f(x) will yield the original x. This is the defining property of inverse functions: they 'undo' the action of the original function.

For example, the natural logarithm function serves as the inverse of the exponential function. Applying the exponential function to a number and then taking the natural logarithm of the result returns the original number, exemplifying the notion that f⁻¹(f(x)) = x. This relationship is fundamental to understanding inverse functions in algebra and calculus.

The statement can also be tested by applying it to familiar operations like addition and multiplication. Subtracting a number and then adding the same number will return to the starting point, just as dividing by a number and then multiplying by the same number will also return to the starting point, highlighting the concept of inverse operations.