📝 When does inverse function exist (14 MCQs)
📖 From Calculus • 1. Basics before calculus • 14 questions available
What is When does inverse function exist?
Definition:
An inverse function exists if and only if the original function is one-to-one (injective), meaning each output comes from exactly one input, which can be tested via the horizontal line test or by showing .
Example:
on all reals is not one-to-one because , so no inverse; but on , it is one-to-one and has inverse .
Reason:
This condition ensures that the inverse is a function itself, avoiding ambiguity when reversing the mapping.
📝 All When does inverse function exist MCQs
Q1. Which of the following best defines a one-to-one (injective) function?
📖 Explanation: A one‑to‑one function never maps two distinct x‑values to the same y‑value. Therefore, whenever x₁≠x₂, we must have f(x₁)≠f(x₂). This property is captured precisely by the statement that different inputs always produce different outputs, which is option B.
Q2. What does the Horizontal Line Test state about a function’s invertibility?
📖 Explanation: The Horizontal Line Test says that a function is invertible exactly when no horizontal line cuts its graph more than once. This ensures that each y‑value comes from at most one x‑value, guaranteeing a well‑defined inverse. Option B restates this condition correctly.
Q3. Consider the piecewise function \f(x)=\\begin{cases}x^{2},&x\\le 0\\\\x+2,&x>0\\end{cases}\. Which statement is true regarding its invertibility?
📖 Explanation: For x≤0, f(x)=x² gives non‑negative outputs; for x>0, f(x)=x+2 yields outputs greater than 2. No y‑value is produced by both pieces, and each piece is itself one‑to‑one. Hence the whole function is injective and passes the Horizontal Line Test, making it invertible. Option B is correct.
Q4. If a function f has an inverse and satisfies \f(2)=5\, which of the following must be true?
📖 Explanation: By definition of an inverse, \f^{-1}(y)\ returns the unique x such that \f(x)=y\. Since \f(2)=5\, the inverse evaluated at 5 must give back 2, i.e., \f^{-1}(5)=2\. The other statements do not follow from the given information. Therefore, option A is correct.
Q5. The function \g(x)=x^{3}-3x\ is defined for all real x. Using the Horizontal Line Test, determine whether g has an inverse.
📖 Explanation: Setting \g(x)=0\ gives \x(x^{2}-3)=0\, whose solutions are \x=0,\\pm\\sqrt{3}\. Thus the horizontal line y=0 intersects the graph at three distinct points, violating the Horizontal Line Test. Consequently, g is not one‑to‑one and has no inverse. Option A is correct.
Q6. Which statement correctly compares the invertibility of \f(x)=x^{2}\ and \h(x)=x^{3}\ on the entire real line?
📖 Explanation: The cubic function \x^{3}\ passes the Horizontal Line Test: each horizontal line meets its graph exactly once, so it is invertible. The quadratic \x^{2}\ fails the test because lines with y>0 intersect twice (e.g., y=4 at x=±2). Hence only h is invertible. Option C is correct.
Q7. Suppose f and g are functions such that the composition \f\\circ g\ is invertible. Which statement must be true?
📖 Explanation: For \f\\circ g\ to be bijective, g must be injective (otherwise the composition would map distinct inputs to the same output) and f must be surjective onto the range of the composition. Neither function needs to be invertible on its own, but both must satisfy the stated conditions. Option D captures this requirement.
Q8. If a function is strictly increasing on its entire domain, what can be concluded about its invertibility?
📖 Explanation: A strictly increasing function never repeats a y‑value; each horizontal line meets its graph at most once. This satisfies the Horizontal Line Test, guaranteeing the existence of a unique inverse function. Hence option A is correct.
Q9. Consider the function \q(x)=x^{2}\ defined on \[-2,2]\. Which domain restriction yields an invertible function?
📖 Explanation: On \[0,2]\ the function \x^{2}\ is monotonic increasing, so each y‑value in \[0,4]\ corresponds to a single x‑value. The restriction to the non‑negative side eliminates the duplicate outputs caused by symmetry about the y‑axis, thereby satisfying the Horizontal Line Test. Option B is correct.
Q10. Why does the function \f(x)=|x|\ not have an inverse, while its restriction to \x\\ge 0\ does?
📖 Explanation: The absolute‑value function yields the same output for x and –x (e.g., \|3|=|-3|=3\), so it is not injective and fails the Horizontal Line Test. When the domain is limited to non‑negative numbers, each y is produced by exactly one x, making the function one‑to‑one and invertible. Option B explains this.
Q11. For the function \f(x)=\\dfrac{x}{1+x^{2}}\, on which intervals is it one‑to‑one?
📖 Explanation: The derivative is \f'(x)=\\dfrac{1-x^{2}}{(1+x^{2})^{2}}\. It is positive for |x|<1 and negative for |x|>1, so f increases on \(-1,1)\ and decreases on \(-\\infty,-1]\ and \[1,\\infty)\. Each of these monotonic intervals is injective; option A lists two of them correctly.
Q12. If a differentiable function passes the Horizontal Line Test, what can be said about its derivative where it exists?
📖 Explanation: A function can be monotonic and still have horizontal tangents (derivative zero) at isolated points, as seen with \x^{3}\ at the origin. The Horizontal Line Test ensures injectivity, which only requires the function to be monotonic, not that its derivative be strictly non‑zero. Thus option C is accurate.
Q13. For a continuous function \f:\\mathbb{R}\\to\\mathbb{R}\ that satisfies the Intermediate Value Property, which additional condition guarantees that \f\ is invertible?
📖 Explanation: Continuity together with the Intermediate Value Property ensures the function takes every value between any two outputs. To be invertible, it must also be one‑to‑one, which is guaranteed if the function is strictly monotonic (always increasing or always decreasing). Hence, strict monotonicity is the required extra condition. Option A is correct.
Q14. Which statement about the function \f(x)=e^{x}+x\ is correct regarding its invertibility on \\\mathbb{R}\?
📖 Explanation: The derivative of \f\ is \f'(x)=e^{x}+1>0\ for all real x, so the function is strictly increasing everywhere. A strictly increasing continuous function satisfies the Horizontal Line Test, guaranteeing the existence of an inverse on the entire real line. Therefore, option B is correct.