# DBT JRF 2022 — Question 20

Exam: DBT JRF 2022 · Section: A · Topic: Aptitude › Calculus - Functions & Inverse Functions · Difficulty: Medium

## Question

The function $y = f(x) = x^2 - 6$ is plotted along with its inverse function $y = f^{-1}(x)$ , where essentially the x & y axis are interchanged. These two curves will intersect at:

## Options

- **A.** (0, 0)
- **B.** (2, 3)
- **C.** (2, 2) & (3, 3)
- **D.** (-2, -2) & (3, 3)

## Answer

Option D

## Explanation

Finding intersections of $y = x^2 - 6$ and its inverse: A function and its inverse intersect along the line $y = x$ . Solve $x = x^2 - 6$ : $x^2 - x - 6 = 0$ Factorise: $(x - 3)(x + 2) = 0$ $x = 3$ or $x = -2$ Check: $f(3) = 9 - 6 = 3$ ✔; $f(-2) = 4 - 6 = -2$ ✔ Intersection points: $(-2, -2)$ and $(3, 3)$ . Note: $x = -2$ is valid only if it lies in the domain of the inverse function. Since $f^{-1}$ is defined for $y \geq -6$ (as $y = x^2 - 6$ has minimum −6), $(-2, -2)$ is valid. Answer: $(-2, -2)$ and $(3, 3)$ (Option 4).

Source: https://mypyqbuddy.com/questions/the-function-y-f-x-x-2---6a104cd5920734408bc0c6f1