Question

. Consider the real function f(x) =x+2/x-2
a) Find the domain and range of the function.
b) Prove that f(x). f (-x) + f (0) = 0​

Answer 1

a) Find the domain and range of the function.

First we will find domain of the given function :-

$\rm \: f(x) = \dfrac{x + 2}{x - 2}$

$\rm x - 2 \neq \: 0 \\ \rm x\neq2$

[ This means that the value of x can be any real number except 2. ]

$\rm \therefore D_f = x\in R - \{2 \} \\ \\ \rm( \: D_f = domain \: of \: the \: given \: function)$

R = real numbers

Now , we will find range of the function :-

Let f(x) be y

$\rm \: f(x) = \dfrac{x + 2}{x - 2} \\ \\ \rm y= \dfrac{x + 2}{x - 2}$

$\rm \rightarrow y({x - 2} )= {x + 2} \\ \\ \rm \rightarrow yx - 2y= {x + 2}\\ \\ \rm \rightarrow yx - x= { 2} + 2y\\ \\ \rm \rightarrow (y - 1)x= { 2} + 2y\\ \\ \rm \rightarrow x= \dfrac{{ 2} + 2y}{y - 1}$

$\rm y - 1 \neq \: 0 \\ \\ \rm y \neq \: 1 \\ \\ \rm R_f = y \in R - \{1 \} \\ \\ \rm R_f =range \: of \: the \: given \: function$

[ This means that the value of y can be any real number except 1. ]

b) Prove that f(x). f (-x) + f (0) = 0

$\rm \rightarrow f(x) = \dfrac{x + 2}{x - 2}$

$\rm \: f( - x) = \dfrac{ - x + 2}{ - x - 2} \\ \\ \rm \rightarrow f( - x) = \dfrac{x - 2}{x + 2}$

$\rm \: f(0) = \dfrac{0 + 2}{0 - 2} \\ \\ \rm f(0) = \dfrac{ 2}{ - 2} \\ \\ \rm \rightarrow f(0) = - 1$

$\rm \longrightarrow f(x). f (-x) + f (0) = 0$

$\rm \longrightarrow \bigg(\dfrac{x + 2}{x - 2} \times \dfrac{x - 2}{x + 2} \bigg) + ( - 1) \\ \\ \rm \longrightarrow 1 - 1 = 0$

Hence proved

Answer 2

Given= Consider the real function f(x) =x+2/x-2

Solution:

Consider the function, f(x) = |x - 2| + |x - 5|, x ∈ R. Statement I f'(4) = 0 Statement II f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5). (a) Statement I is false, Statement II is true (b) Statement II is true, Statement II is true; Statement II is a correct explanation of Statement I (c) Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I (d) Statement I is true, Statement II is false

Given= Find the domain and range of the function.

Solution:

The domain of a function is the set of all possible inputs for the function. For example, the domain of f(x)=x² is all real numbers, and the domain of g(x)=1/x is all real numbers except for x=0. We can also define special functions whose domains are more limited.

The domain of a function is the set of all acceptable input values (X-values). The range of a function is the set of all output values (Y-values).

Question= Prove that f(x). f (-x) + f (0) = 0

Solution is given in above pic⤴️