Question

Select all of the following functions which are invertible.

f(x) = x ^ 2 + 4
f(x) = (x + 5) ^ 3
f(x) = 3/8 * x + 7
f(x) = 73

Answer 1

From the given list, the invertible functions are f(x) = (x + 5) ^ 3 and f(x) = 3/8 * x + 7 since they are both one-to-one functions. The function f(x) = x ^ 2 + 4 is not invertible over the real numbers, and f(x) = 73 is not invertible because it is a constant function.

Step-by-step explanation:

To determine which functions are invertible, we look for functions that are one-to-one, where each input has one distinct output. This property ensures that the inverse function will also be well-defined.

• f(x) = x ^ 2 + 4 is not invertible on the set of all real numbers because it fails the horizontal line test. Squaring a number results in losing the sign information, so for each positive output, there are two inputs (one positive and one negative).
• f(x) = (x + 5) ^ 3 is invertible because for a cubic function, each input maps to a unique output, satisfying the one-to-one requirement.
• f(x) = 3/8 * x + 7 is invertible. This is a linear function with a non-zero slope, which means it is one-to-one and can be inverted by solving for x in terms of y.
• f(x) = 73 is not invertible because it is a constant function, which maps every input to the same output, violating the one-to-one criterion.

Therefore, the invertible functions from the given list are f(x) = (x + 5) ^ 3 and f(x) = 3/8 * x + 7.