If f is a real function satisfying f x -1- x - x 2-1- x 2 for all x - R -0- then write the expression for f x -

We have, f ( x+1/x )=x^2+1/x^2 Now, =x^2+1/x^2 = ( x+1/x )^2 2 [∵ (a+b)^2 = a^2+b^2+2ab] ⇒ f ( x+1/x ) = ( x+1/x )^2 2 ⇒ f(x) =x^2 2, where |x| ≥ 2 ...

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Standard XII

Mathematics

Real Valued Functions

If f is a rea...

Question

If f is a real function satisfying $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$ for all $x ϵ R - {0}$ , then write the expression for f(x).

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Solution

We have,
$f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$
Now,
$= x^{2} + \frac{1}{x^{2}} = {(x + \frac{1}{x})}^{2} - 2$
$[∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b]$
$\Rightarrow f (x + \frac{1}{x}) = {(x + \frac{1}{x})}^{2} - 2 \Rightarrow f (x) = x^{2} - 2, w h e r e | x | \geq 2$

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If f is a real function satisfying f(x+1x)=x2+1x2 for all xϵR−{0}, then write the expression for f(x).

We have, f(x+1x)=x2+1x2 Now, =x2+1x2=(x+1x)2−2 [∵(a+b)2=a2+b2+2ab] ⇒f(x+1x)=(x+1x)2−2⇒f(x)=x2−2, where |x|≥2

If f is a real function satisfying $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$ for all $x ϵ R - {0}$ , then write the expression for f(x).

We have,
$f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$
Now,
$= x^{2} + \frac{1}{x^{2}} = {(x + \frac{1}{x})}^{2} - 2$
$[∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b]$
$\Rightarrow f (x + \frac{1}{x}) = {(x + \frac{1}{x})}^{2} - 2 \Rightarrow f (x) = x^{2} - 2, w h e r e | x | \geq 2$