Inverse functions

Answer

f⁻¹ undoes the effect of f — it reverses the mapping. If f maps a → b, then f⁻¹ maps b → a. Together: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x.

Answer

f(f⁻¹(x)) = x (applying f after f⁻¹ gives back x). f⁻¹(f(x)) = x (applying f⁻¹ after f gives back x). Both compositions return the original input — they cancel each other out.

Answer

f⁻¹ reverses the mapping: f⁻¹(8) = 3 and f⁻¹(12) = 5. No formula needed — just swap the input and output of f.

Answer

f⁻¹(x) is the inverse function, not the reciprocal. 1/f(x) means "1 divided by the output of f" — a completely different thing. The −1 in f⁻¹ is function notation for "inverse," not an exponent.

Answer

1. Write y = f(x). 2. Swap x and y (write x = f(y)). 3. Rearrange to make y the subject. 4. Replace y with f⁻¹(x).

Answer

y = 4x − 7. Swap: x = 4y − 7. Rearrange: x + 7 = 4y → y = (x + 7)/4. f⁻¹(x) = (x + 7)/4.

Answer

y = (2x + 1)/3. Swap: x = (2y + 1)/3. Rearrange: 3x = 2y + 1 → 2y = 3x − 1 → y = (3x − 1)/2. f⁻¹(x) = (3x − 1)/2.

Answer

They will get x = (expression in y), not y = (expression in x). The swap is essential — it converts the input-output relationship. Without swapping, the result is not expressed as f⁻¹(x).

Answer

An inverse only exists if f is one-to-one (each output comes from exactly one input). Example: f(x) = x² over all ℝ is not one-to-one — f(3) = f(−3) = 9, so the inverse would give two outputs. Restricting to x ≥ 0 makes it one-to-one: f⁻¹(x) = √x.

Answer

y = x². Swap: x = y². Rearrange: y = √x (take positive root since original domain x ≥ 0). f⁻¹(x) = √x, domain x ≥ 0.

Answer

The domain of f⁻¹ equals the range of f. The range of f⁻¹ equals the domain of f. The inverse swaps domain and range — inputs become outputs and vice versa.

Answer

Without a domain restriction, f(x) = x² is not one-to-one — the inverse is not unique. The full answer must be: f⁻¹(x) = √x for x ≥ 0. IB questions typically award a separate mark for correctly stating the domain of f⁻¹.

Answer

The graph of f⁻¹ is the reflection of the graph of f in the line y = x. Every point (a, b) on f maps to the point (b, a) on f⁻¹ — x and y coordinates are swapped.

Answer

(7, 2) and (4, −1). The inverse swaps x and y — every (a, b) on f becomes (b, a) on f⁻¹.

Answer

At any intersection point, f(x) = f⁻¹(x). These points also lie on the line y = x (since they satisfy f(x) = x at the intersection in the most common case). Note: f and f⁻¹ can intersect off the line y = x too, but they always cross y = x when they intersect.

Answer

The correct reflection is over the line y = x, not the x-axis. Reflecting over the x-axis would flip the graph vertically — that gives −f(x), not f⁻¹(x). The line y = x is the mirror that swaps x and y coordinates.