Let $f : \bullet \to \bullet $ satisfy $f (x) f ( y) = f (xy) $ for all real numbers $x$ and $y$. If $f (2)= 4$, then $f \left( \frac{1}{2} \right) = $
- 0
- $\frac{1}{4}$
- $\frac{1}{2}$
- 1
The Correct Option is B
Solution and Explanation
Top Questions on Functions
- If the domain of the function \(f(x)\) = \(\frac{\sqrt{x^2-25}}{(4-x^2)}+ \log(x^2+2x-15)\) is \((-∞, α) ∪ (β, ∞)\), then \(α^2+β^2\) is equal to \(b\)
- Let the function \(f:\R→\R\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}}\frac{(x)^{2023}+2024x+2025}{(x^2-x+3)}+\frac{2}{e^{\pi x}}\frac{(x^{2023}+2024x+2025)}{(x^2-x+3)}\)
Then the number of solutions of f (x) = 0 in \(\R\) is __. - Let \(f:[0,\frac{\pi}{2}]→[0,1]\) be the function defined by \(f(x)=\sin^2x\) and let \(g:[0,\frac{\pi}{2}]→[0,\infin)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}=x^2}\).
- Let \(f:\R→\R\) be a function such that f(x + y) = f(x) + f(y) for all x, y ∈ \(\R\), and \(g:\R→(0,\infin)\) be a function such that g(x + y) = g(x)g(y) for all x, y ∈ \(\R\). If \(f(\frac{-3}{5})=12\) and \(g(\frac{-1}{3})=2\), then the value of \((f(\frac{1}{4})+g(-2)-8)g(0)\) is ______.
- Let \(f:\R→\R\) be a function defined by
\(f(x) = \begin{cases} x^2\sin(\frac{\pi}{x^2}), & \text{if } x \ne 0, \\ 0, & \text{if } x = 0. \end{cases}\)
Then which of the following statements is TRUE ?
Questions Asked in KEAM exam
- Let \(k\) be a real number such that \(\sin \dfrac{3π}{14} \cos \dfrac{3π}{14} = k \cos \dfrac{π}{14}\).Then the value of \(4k\) is
- KEAM - 2023
- Trigonometry
- Suppose \(A=\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{bmatrix}\) is an adjoint of the matrix \(\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}\). Then the value of \(\frac{a_!+b_2+c_3}{b_1a_2}\) is
- KEAM - 2023
- Matrices
- Let \(f:R→R\) be a function defined by
\(f(x) = \begin{cases} 3e^x & \quad \text {if}\ {x<0}\\ x^2+3x+3 & \quad \text {if}\ 0≤x<1 \\ x^2-3x-3 & \quad \text {if} \ x≥1\end{cases}\)- KEAM - 2023
- Limit and Continuity
The angle of minimum deviation for a prism of apex angle 60° and refractive index of \(\sqrt{2}\) is:
- KEAM - 2023
- Ray optics and optical instruments
- A thin particle moves from \((0,1)\) and gets reflected upon hitting the x-axis at \((√3,0)\). Then the slope of the reflected line is ?
- KEAM - 2023
- Slope of a line
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
SUBSCRIBE TO OUR NEWS LETTER
