Questions: Inverse Functions Review

Question 1

If f(3) = 7, what is f⁻¹(7)?

• A. 1/7 — since f⁻¹ means the reciprocal of the output
• B. 3 — since the inverse reverses the input-output pair
• C. 1/3 — since the inverse undoes multiplication by 3
• D. Cannot be determined without knowing the formula for f

Answer: (B)

The inverse function reverses input and output: f(3) = 7 means f⁻¹(7) = 3 by definition. You don't need the formula — just the input-output relationship. Option A is the classic confusion between f⁻¹(x) (inverse function) and [f(x)]⁻¹ = 1/f(x) (reciprocal). These are completely different things in function notation.

Question 2

The function f(x) = x² is defined over all real numbers. Which statement is true?

• A. f⁻¹(x) = √x for all real x, since squaring and square-rooting are inverse operations
• B. f has no inverse over all reals, because f(2) = f(−2) = 4 — two inputs produce the same output
• C. f⁻¹(x) = −√x recovers the original input for all real x
• D. f has an inverse because every output in the range has at least one corresponding input

Answer: (B)

Since f(2) = f(−2) = 4, f is not one-to-one over all reals — the horizontal line y = 4 hits the parabola twice. The inverse can't exist because it can't decide whether to return 2 or −2 for the input 4. Option A sounds reasonable but ignores this non-uniqueness. To get a valid inverse, you must restrict the domain to x ≥ 0 (giving f⁻¹(x) = √x) or x ≤ 0 (giving f⁻¹(x) = −√x), but not both simultaneously.

Question 3

If (5, 12) is a point on the graph of f, then (12, 5) is a point on the graph of f⁻¹.

• T. True
• F. False

Answer: True

Inverse functions swap the roles of input and output: f(5) = 12 implies f⁻¹(12) = 5 by definition. Graphically, this corresponds to reflecting the point (5, 12) across the line y = x to get (12, 5). This coordinate swap is the defining geometric property of the inverse — which is why the graphs of f and f⁻¹ are always reflections of each other across y = x.

Question 4

The notation f⁻¹(x) means the same as [f(x)]⁻¹ = 1/f(x).

• T. True
• F. False

Answer: False

The −1 superscript in function notation denotes the inverse function, not the reciprocal. f⁻¹(x) is the function that undoes f, satisfying f⁻¹(f(x)) = x. For example, if f(x) = 2x, then f⁻¹(x) = x/2, while 1/f(x) = 1/(2x) — completely different functions. The notation is confusingly similar to the reciprocal exponent in arithmetic, but in function composition context, f⁻¹ always means inverse, not reciprocal.

Question 5

Why must a function be one-to-one to have an inverse, and what happens if it isn't?

Model answer: A function must be one-to-one because the inverse must assign a unique output to each input. If two inputs a ≠ b both map to the same output (f(a) = f(b)), the inverse can't decide which input to return — it would need to produce two outputs for one input, violating the definition of a function. If a function isn't one-to-one over its full domain, you can still define a partial inverse by restricting the domain to a region where it is one-to-one, as with restricting x² to x ≥ 0 to define √x.

The inverse function is itself a function, so it must pass the vertical line test — each input maps to exactly one output. Non-one-to-one functions create ambiguity: the inverse at an output value c doesn't know which of the multiple preimages to return. Domain restriction resolves this by discarding the ambiguous inputs. This is exactly why arcsin is defined only on [−π/2, π/2] — sin is not one-to-one over all reals, so you pick the interval where it is.