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# Let $f\;:\;R\rightarrow R$ be the function defined by $f(x)=\frac{1}{2-\cos x}\forall x\;\in R.$Then,find the range of f.

Toolbox:
• Range of f is the possible value f(x) can take for all $x \in R$
• $\cos x$ can take values from -1 to1 only
Given $f:R \to R$
$f(x)=\frac{1}{2 -\cos x} \qquad x \in R$
$\cos x$ can take values between -1 and 1 only
When $\cos x =-1$
$f(x)=\frac{1}{2-1(-1)}=\frac{1}{3}$
When $\cos x=1$
$f(x)=\frac{1}{2-1}=\frac{1}{1}=1$
Solution:Range of $f=[\frac{1}{3},1]$
edited Mar 30, 2013