Maths. Section B1Relations and Functions

Let f(x)=cos^-1≤ft((x^2)/(1+x^2)). The range of f is

Let $f(x)=\cos^{-1}\left(\frac{x^2}{1+x^2}\right)$. The range of f is

Answer & explanation

Correct answer: option 4

$\left(\frac{x^2}{1+x^2}\right)≤1$

This is true for all x ∈ R. So, the domain = R

Now, $\frac{x^2}{1+x^2}=1-\frac{1}{1+x^2}$

$∴ 0≤\frac{x^2}{1+x^2}<1;\cos^{-1}0≥\cos^{-1}\frac{x^2}{1+x^2}>\cos^{-1}1⇒\frac{π}{2}≥\cos^{-1}\frac{x^2}{1+x^2}>0$

∴ the range = $\left(0,\frac{π}{2}\right]$