Question

Let f be a function from Z to Z. such that f(x) = x + 3 Find the inverse of f?

• A. x -3
• B. x + 3
• C. x - 6
• D. x - 9

Answer 1

Answer:

A. x -3

Understanding Inverse Functions

• An inverse function, denoted as f⁻¹(x), "undoes" the operation of the original function f(x). If f(a) = b, then f⁻¹(b) = a.

• For a function to have an inverse, it must be bijective. This means the function must be both:

  • One-to-one (Injective): Each distinct element in the domain maps to a distinct element in the codomain. No two different inputs produce the same output.

  • Onto (Surjective): Every element in the codomain is mapped to by at least one element in the domain. The range of the function is equal to its codomain.

• For the given function f(x) = x + 3, where f: Z → Z (from integers to integers):

  • It is one-to-one because if x₁ + 3 = x₂ + 3, then x₁ = x₂.

  • It is onto because for any integer y in the codomain, we can find an integer x = y - 3 in the domain such that f(x) = (y - 3) + 3 = y.

  • Since it is both one-to-one and onto, an inverse function exists.

Steps to Find the Inverse of a Function

1. Replace f(x) with y: Start by writing the function as y = f(x). For f(x) = x + 3, this becomes y = x + 3.

2. Swap x and y: Interchange the variables x and y in the equation. This represents the reflection of the function across the line y = x, which is the geometrical interpretation of an inverse. So, x = y + 3.

3. Solve for y: Isolate y in the new equation. This will give you the expression for the inverse function. From x = y + 3, subtract 3 from both sides to get y = x - 3.

4. Replace y with f⁻¹(x): The expression you found for y is the inverse function. So, f⁻¹(x) = x - 3.