Statement -1 : Any function f (x) is even function, when f (-x) =… - Vidyadip

MCQ

Statement $-1$ : Any function $f (x)$ is even function, when $f (-x) = f (x)$ over its specified domain.

Statement $-2$ : $f(x) = \frac{1}{{\sqrt {1 - {x^2}} }} + \left[ {\frac{{{x^2} + x + 1}}{4}} \right]$ , where $[.]$ is greatest integer function. Function $f(x)$ is even function

A
Statement $-1$ is true, statement $-2$ is true but statement $-1$ is not the correct explanation for statement $-2$
B
Statement $-1$ is true, statement $-2$ is false
C
Statement $-1$ is false, statement $-2$ is true
✓
Both statements are true, and statement $-1$ is the true explanation of statement $-2$

✓

Answer

Correct option: D.

Both statements are true, and statement $-1$ is the true explanation of statement $-2$

d
$f(x)=\frac{1}{\sqrt{1-x^{2}}}+\left[\frac{x^{2}+x+1}{4}\right]$

domain of $f(x)$ is $x \in(-1,1)$

within that domain, $\left[\frac{x^{2}+x+1}{4}\right]=0$

So,

$f(x)=\frac{1}{\sqrt{1-x^{2}}}$ i.e, an even function.

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