Let f(x)=\(\begin{Bmatrix} x+1&\,\,\,-1\leq x\leq 0\\ -x&\,\,\,\,\,0<x\leq1 \end{Bmatrix}\)
- f(x) is discontinuous in [-1,1] and so has no maximum value or minimum value in [-1,1].
- f(x) is continuous in [-1,1] and so has maximum and minimum value
- f(x) is discontinuous in [-1,1] but still has the maximum and minimum value
- f(x) is bounded in[-1,1] and doesn't attain maximum or minimum value
The Correct Option is D
Solution and Explanation
The function f(x) is bounded within the interval [-1, 1], and it does not attain a maximum or minimum value within this interval.
The correct option is(D): f(x) is bounded in[-1,1] and doesn't attain maximum or minimum value
Top Questions on Functions
- The sum of all local minimum values of the function \( f(x) \) as defined below is:
\[ f(x) = \begin{cases} 1 - 2x & \text{if } x < -1, \\[10pt] \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2, \\[10pt] \frac{11}{18}(x-4)(x-5) & \text{if } x > 2. \end{cases} \] - In \( I(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} \, dx \), where \( m, n > 0 \), then \( I(9, 14) + I(10, 13) \) is:
- Let \( f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32} \). Then the value of \[ 8 \left( f\left( \frac{1}{15} \right) + f\left( \frac{2}{15} \right) + \dots + f\left( \frac{59}{15} \right) \right) \] is equal to:
- 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:
- The area of the region enclosed by the curves \( y = e^x \), \( y = |e^x - 1| \), and the y-axis is:
Questions Asked in WBJEE exam
- Let \(f : \mathbb{R} \to \mathbb{R}\) be given by \(f(x) = |x^2 - 1|\), then:
- \(f(x) = \cos x - 1 + \frac{x^2}{2!}, \, x \in \mathbb{R}\)
Then \(f(x)\) is: - Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function and \(f(1) = 4\). Then the value of
\[ \lim_{x \to 1} \int_{4}^{f(x)} \frac{2t}{x - 1} \, dt \]- WBJEE - 2024
- Integration
- If \(\int \frac{\log(x + \sqrt{1 + x^2})}{1 + x^2} \, dx = f(g(x)) + c\), then:
- WBJEE - 2024
- Integration
- The equation \(2x^5 + 5x = 3x^3 + 4x^4\) has:
- WBJEE - 2024
- Quadratic Equation
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