Let f - R - R be the function defined by f x -1-2-cos x- x - R-Then- find the range of f-

Given function, f(x)=1/2 cos x,∀ x ∈ R Let y =1/2 cos x ⇒ 2y y cos x=1 ⇒ y cos x =2y 1 ⇒ cos x =2y 1/y ⇒ cos x =2 1/y Now, 1 ≤ cos x ≤ 1 ⇒ 1 ≤ 2 1/ ...

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Standard XII

Mathematics

Domain and Range of Trigonometric Ratios

Let f : R → R...

Question

Let $f : R \to R$ be the function defined by $f (x) = \frac{1}{2 - c o s x^{'}} \forall x \in R$ .Then, find the range of f.

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Solution

Given function, $f (x) = \frac{1}{2 - c o s x}, \forall x \in R$
Let $y = \frac{1}{2 - c o s x}$
$\Rightarrow 2 y - y c o s x = 1 \Rightarrow y c o s x = 2 y - 1 \Rightarrow c o s x = \frac{2 y - 1}{y} \Rightarrow c o s x = 2 - \frac{1}{y} N o w, - 1 \leq c o s x \leq 1 \Rightarrow - 1 \leq 2 - \frac{1}{y} \geq 1 \Rightarrow - 3 \leq - \frac{1}{y} \leq - 1 \Rightarrow 1 \leq \frac{1}{y} \leq 3 \Rightarrow \frac{1}{3} \leq y \leq 1$
So, the range of f is $[\frac{1}{3}, 1]$

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