Question

Select the function that does not have a well-defined inverse.

1) f: → f(x)=ŒˆxŒ‰
2) f: → f(x)=3x⁴
3) f: → f(x)=Œˆx²Œ‰
4) f: → f(x)=-2x⁵

Answer 1

Function 3, f(x) = x², does not have a well-defined inverse because it is not a one-to-one function. Both positive and negative values of x produce the same positive result for f(x), which prevents the function from having a unique inverse.

Step-by-step explanation:

We are looking for a function that does not have a well-defined inverse. An inverse function effectively "undoes" the original function. For instance, the square root is the inverse of squaring a number, and the natural logarithm is the inverse of the exponential function ex. For a function to have an inverse, it must be bijective—both one-to-one (injective) and onto (surjective). Upon examining the given functions, let's consider their properties:

• f(x) = x4 is not one-to-one because both positive and negative inputs yield the same positive result.
• f(x) = x2 also is not one-to-one for the same reason.
• f(x) = -2x5 is an odd function and is one-to-one.

Function 3, f(x) = x2, does not have a well-defined inverse because it is not one-to-one. For every positive value of f(x), there are two possible values of x (one positive and one negative) that could have produced that output result. This characteristic violates the requirement for a function to be one-to-one (injective) in order to have an inverse function.