If f is an even function defined on interval (-5, 5), then four real values of x satisfying equation

English

Gujarati

Hindi

1.Relation and Function

hard

If $f$ is an even function defined on the interval $(-5, 5),$ then four real values of $x$ satisfying the equation $f(x) = f\left( {\frac{{x + 1}}{{x + 2}}} \right)$ are

$\frac{{ - 3 - \sqrt 5 }}{2},\;\frac{{ - 3 + \sqrt 5 }}{2},\;\frac{{3 - \sqrt 5 }}{2},\;\frac{{3 + \sqrt 5 }}{2}$

$\frac{{ - 5 + \sqrt 3 }}{2},\;\frac{{ - 3 + \sqrt 5 }}{2},\;\frac{{3 + \sqrt 5 }}{2},\;\frac{{3 - \sqrt 5 }}{2}$

$\frac{{3 - \sqrt 5 }}{2},\;\frac{{3 + \sqrt 5 }}{2},\;\frac{{ - 3 - \sqrt 5 }}{2},\;\frac{{5 + \sqrt 3 }}{2}$

$ - 3 - \sqrt 5 ,\; - 3 + \sqrt 5 ,\;3 - \sqrt 5 ,\;3 + \sqrt 5 $

(IIT-1996)

Solution

(a) Since $f$ is an even function $f\,( – x) = f(x)\,,\forall x \in ( – 5,\,5).$

We are given that $f(x) = f\,\left( {\frac{{x + 1}}{{x + 2}}} \right)$

$ \Rightarrow \,\,f( – x) = f\left( {\frac{{ – x + 1}}{{ – x + 2}}} \right)\, \Rightarrow \,\,f(x) = f\,\left( {\frac{{ – x + 1}}{{ – x + 2}}} \right)$

To find the values of $x$, we set

$x = \frac{{ – x + 1}}{{ – x + 2}}\,\, \Rightarrow \,\,x = \frac{{3 \pm \sqrt {9 – 4} }}{2} = \frac{{3 \pm \sqrt 5 }}{2}$

Also $f(x) = f\,\left( {\frac{{x + 1}}{{x + 2}}} \right) = f( – x)$

To find the values of $x$, we set

$ – x = \frac{{x + 1}}{{x + 2}}\,\, \Rightarrow \,\,x = \frac{{ – 3 \pm \sqrt {9 – 4} }}{2} = \frac{{ – 3 \pm \sqrt 5 }}{2}$

Thus the four required values of $x$ are

$\frac{{ – 3 – \sqrt 5 }}{2},\,\frac{{ – 3 + \sqrt 5 }}{2},\,\frac{{3 – \sqrt 5 }}{2},\,\frac{{3 + \sqrt 5 }}{2}.$

Standard 12

Mathematics

If ${e^x} = y + \sqrt {1 + {y^2}} $, then $y =$

medium

Let, $f(x)=\left\{\begin{array}{l} x \sin \left(\frac{1}{x}\right) \text { when } x \neq 0 \\ 1 \text { when } x=0 \end{array}\right\}$ and $A=\{x \in R: f(x)=1\} .$ Then, $A$ has

hard

(KVPY-2019)

Which of the following is function

normal

Consider a function $f:\left[ { – 1,1} \right] \to R$ where $f(x) = {\alpha _1}{\sin ^{ – 1}}x + {\alpha _3}\left( {{{\sin }^{ – 1}}{x^3}} \right) + ….. + {\alpha _{(2n + 1)}}{({\sin ^{ – 1}}x)^{(2n + 1)}} – {\cot ^{ – 1}}x$ Where $\alpha _i\ 's$ are positive constants and $n \in N < 100$ , then $f(x)$ is

normal

Suppose $f$ is a function satisfying $f ( x + y )= f ( x )+ f ( y )$ for all $x , y \in N$ and $f (1)=\frac{1}{5}$. If $\sum \limits_{n=1}^m \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$, then $m$ is equal to $……………$.

hard

(JEE MAIN-2023)

If $f$ is an even function defined on the interval $(-5, 5),$ then four real values of $x$ satisfying the equation $f(x) = f\left( {\frac{{x + 1}}{{x + 2}}} \right)$ are

Solution

Similar Questions

If ${e^x} = y + \sqrt {1 + {y^2}} $, then $y =$

Let, $f(x)=\left\{\begin{array}{l} x \sin \left(\frac{1}{x}\right) \text { when } x \neq 0 \\ 1 \text { when } x=0 \end{array}\right\}$ and $A=\{x \in R: f(x)=1\} .$ Then, $A$ has

Which of the following is function

Suppose $f$ is a function satisfying $f ( x + y )= f ( x )+ f ( y )$ for all $x , y \in N$ and $f (1)=\frac{1}{5}$. If $\sum \limits_{n=1}^m \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$, then $m$ is equal to $……………$.

Start a Free Trial Now