Let f \left(\right. \frac{- \pi}{2} , \textrm{ } \frac{\pi}{2} \left.\right) : \rightarrow R be given by f(x) = \left(\left(\right. \textrm{ } log \textrm{ } \left(\right. secx + tanx \left.\right) \textrm{ } \left.\right)\right)^{\textrm{ } 3}. Then

Engineering

Mathematics

Classification of Functions

Even and Odd Functions

Question

Let f $(\frac{- π}{2}, \frac{π}{2}) : \to$ R be given by f(x) = ${(\log (secx + tanx))}^{3}$ .

Then

f(x) is an even function

f(x) is an odd function

f(x) is a one‑one function

f(x) is an onto function

LiveFree JEE Main Previous Year Online Papers Live Free JEE Advance Previous Year Online Papers

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

f(x) = – f (– x)

odd function

f '(x) = 3 (log (sec x + tan x))² (sec x tan x + sec²x)

= 3 (log (sec x + tan x))² (sec x) [sec x + tan x]

f '(x) > 0 in x $\in (\frac{- π}{2}, \frac{π}{2})$

say t = sec x + tan x $(\frac{1 + sinx}{cosx}) = \frac{\sin \frac{x}{2} + \cos \frac{x}{2}}{\cos \frac{x}{2} - \sin \frac{x}{2}} = \frac{1 + \tan \frac{x}{2}}{1 - \tan \frac{x}{2}}$

$\Rightarrow T = \tan (\frac{π}{4} + \frac{x}{2})$

$x \in (\frac{- π}{2}, \frac{π}{2}) \Rightarrow x \in (\frac{- π}{4}, \frac{π}{4}) \Rightarrow \frac{π}{4} + \frac{x}{2} \in (0, \frac{π}{2})$

$∴ t \in (0, \infty) \Rightarrow$ y = (log t)³ $\in (- \infty, \infty)$

$∴$ Onto function.