Let \(R\) be the set of all real numbers and \(f : R \rightarrow R\) be given by \(f(x) = 3x^2 + 1\).Then the set \(f ^{-1}\left(\left[1, 6\right]\right)\) is
- $\left\{-\sqrt{\frac{5}{3}},0, \sqrt{\frac{5}{3}}\right\}$
- $\left[-\sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right]$
- $\left[-\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right]$
- $\left(-\sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right)$
The Correct Option is B
Solution and Explanation
Given, \(y=3 x^{2}+1\)
\(\Rightarrow 3 x^{2} =y-1\)
\(\Rightarrow x^{2}=\frac{y-1}{3}\)
\(\Rightarrow x=\pm \sqrt{\frac{y-1}{3}}\)
\(\therefore f^{-1}(x)=\pm \sqrt{\frac{x-1}{3}}\)
When \(x \in[1,6]\)
Then, \(f^{-1}(x) \in\left[-\sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right]\)
Top Questions on Functions
- If \( y = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x) \), where \(\operatorname{sgn}(p)\) denotes the signum function of \(p\), then the sum of elements in the range of \(y\) is:
- Statement 1 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one–one.
Statement 2 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many–one.
Which of the following is correct? - If domain of \(f(x) = \sin^{-1}\left(\frac{5-x}{2x+3}\right) + \frac{1}{\log_{e}(10-x)}\) is \((-\infty, \alpha] \cup (\beta, \gamma) - \{\delta\}\) then value of \(6(\alpha + \beta + \gamma + \delta)\) is equal to :
- Assertion (A): Let \( A = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \). If \( f : A \to A \) be defined as \( f(x) = x^2 \), then \( f \) is not an onto function.
Reason (R): If \( y = -1 \in A \), then \( x = \pm \sqrt{-1} \notin A \). - If $ f(x) = 2x^2 - 3x + 5 $, find $ f(3) $.
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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