Let f,g : R → R be functions defined by*
\(f(x) = \begin{cases} [x], & x < 0 \\ |1 - x|, & x \geq 0 \end{cases}\)
and \(g(x) = \begin{cases} e^x - x, & x < 0 \\ {(x - 1)^2 - 1}, & x \geq 0 \end{cases}\)
Where [x] denotes the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly:
Let f,g : R → R be functions defined by*
\(f(x) = \begin{cases} [x], & x < 0 \\ |1 - x|, & x \geq 0 \end{cases}\)
and \(g(x) = \begin{cases} e^x - x, & x < 0 \\ {(x - 1)^2 - 1}, & x \geq 0 \end{cases}\)
Where [x] denotes the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly:
- one point
- two points
- three points
- four points
The Correct Option is B
Solution and Explanation
The correct answer is (B) : two points
\(f(x) = \begin{cases} [x], & x < 0 \\ |1 - x|, & x \geq 0 \end{cases}\) and \(g(x) = \begin{cases} e^x - x, & x < 0 \\ {(x - 1)^2 - 1}, & x \geq 0 \end{cases}\)
\((fog)(x) = \begin{cases} [g(x)], & g(x) < 0 \\ 1 - g(x), & g(x) \geq 0 \end{cases}\)
\(h(x) = \begin{cases} |1 + x - e^x|, & x < 0 \\ 1, & x = 0 \\ (x - 1)^2 - 1, & 0 < x < 2 \\ |2 - (x - 1)^2|, & x \geq 2 \end{cases}\)
So, x = 0, 2 are the two points where fog is discontinuous.
Top Questions on Functions
- Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 \). If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of \( 28 \sum_{i=1}^5 f(i) \) is:
- Let the function $ f(x) = \sqrt{\log_e(1 - x^2)} $. Then the domain of $ f(x) $ is:
- The number of points of discontinuity of the function $ f(x) = \left\lfloor \frac{x^2}{2} \right\rfloor - \left\lfloor \sqrt{x} \right\rfloor, \quad x \in [0, 4], $ where $ \left\lfloor \cdot \right\rfloor $ denotes the greatest integer function, is:
- If the domain of the function $ f(x) = \log_e \left( \frac{2x-3}{5+4x} \right) + \sin^{-1} \left( \frac{4+3x}{2-x} \right) $ is $ [\alpha, \beta] $, then $ \alpha^2 + 4\beta $ is equal to
- Let the domain of the function $ f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2) $ be (a, b). If $ \int_0^{a+b} [x^2] dx = p - q\sqrt{r} $, $ p, q, r \in \mathbb{N} $, gcd(p, q, r) = 1, where [.] is the greatest integer function, then p + q + r is equal to
Questions Asked in JEE Main exam
- Density of 3 M NaCl solution is 1.25 g/mL. The molality of the solution is:
- JEE Main - 2025
- Solutions
In the given circuit the sliding contact is pulled outwards such that the electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 Ω, the value of the current in the circuit will be A.- JEE Main - 2025
- Electrical Circuits
- Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k} \). Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in {R} \) and \( L_2 : \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2 \), and is parallel to \( \vec{a} + \vec{b} \), then \( L_3 \) passes through the point:
- JEE Main - 2025
- Vectors
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Relations and functions
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions