Which of the following does NOT have an inverse function?

Question

BlackTom AI · Accepted Answer

A function has an inverse exactly when it is one-to-one (passes the horizontal line test) — each output comes from exactly one input.

1) Relevant concepts
- One-to-one (injective): no horizontal line intersects the graph more than once.
- Monotonicity: strictly increasing or decreasing functions are one-to-one.
- If solving y = f(x) for x produces two possible x-values (±), the inverse is not a single-valued function without domain restriction.

2) Step-by-step reasoning and algebraic checks

Option A: f(x) = sqrt(x+3) − 4, domain x ≥ −3. Let y = sqrt(x+3) − 4.
  y + 4 = sqrt(x+3)  →  (y+4)^2 = x+3  →  x = (y+4)^2 − 3.
This yields a single x for each valid y (because principal square root is used), so f is one-to-one on its domain. Inverse exists.

Option B: f(x) = 1/(x+3) − 4, domain x ≠ −3. Let y = 1/(x+3) − 4.
  y + 4 = 1/(x+3)  →  x+3 = 1/(y+4)  →  x = 1/(y+4) − 3.
Derivative f'(x) = −1/(x+3)^2 < 0 for all x ≠ −3, so f is strictly decreasing on its domain and one-to-one. Inverse exists.

Option C: f(x) = (x+3)^2 − 4, domain all real numbers. Let y = (x+3)^2 − 4.
  y + 4 = (x+3)^2  →  x+3 = ± sqrt(y+4)  →  x = −3 ± sqrt(y+4).
This gives two possible x-values for many y (e.g., f(−2) = 1 − 4 = −3 and f(−4) = 1 − 4 = −3), so f fails the horizontal line test and is not one-to-one on R. Without restricting the domain (e.g., x ≥ −3), no single-valued inverse exists. Therefore this function does NOT have an inverse function.

Option D: f(x) = (x+3)^3 − 4. Let y = (x+3)^3 − 4.
  y + 4 = (x+3)^3  →  x+3 = cuberoot(y+4)  →  x = cuberoot(y+4) − 3.
Cube is strictly increasing on R, so this gives a unique x for each y; inverse exists.

3) Why the correct answer is correct
Option C is the only function that maps two different x-values to the same y (parabola symmetry), so it is not one-to-one and has no inverse without domain restriction.

4) Why other options are incorrect
A is one-to-one on its domain (principal sqrt), B is strictly decreasing and one-to-one, D is strictly increasing (cubic) and one-to-one.

Conclusion: The function without an inverse is C. This matches the provided answer: C.

Which of the following does NOT have an inverse function?单项选择题

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