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Question: How do you simplify \(\dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{{{x}^{4}}-1}\)and state the domain?...

How do you simplify xx2+1÷xx4−1\dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{{{x}^{4}}-1}and state the domain?

Explanation

Solution

To solve this question, firstly simplify the given algebraic expressions using algebraic identities and write them in their factors. Then, see that if any like terms are present and cancel them. After cancelling all the like terms, simply multiply all the terms in the numerator and denominator and see if the fraction can be simplified to get the final answer.

Complete step by step solution:
Given:
The expression given is xx2+1÷xx4−1\dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{{{x}^{4}}-1}…(i)\left( i \right)
The two fractions involved in the expression are xx2+1\dfrac{x}{{{x}^{2}}+1} andxx4−1\dfrac{x}{{{x}^{4}}-1}
Let us first simplify the second fraction using the basic algebraic identity given as a2−b2=(a−b)(a+b){{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)and applying this identity in the denominator of the fraction,
⇒xx4−1=x(x2−1)(x2+1)\Rightarrow \dfrac{x}{{{x}^{4}}-1}=\dfrac{x}{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}+1 \right)} …(ii)\left( ii \right)
Now, by substituting equation (ii)\left( ii \right)back in the equation(i)\left( i \right) , we get,
⇒xx2+1÷xx4−1=xx2+1÷x(x2−1)(x2+1)\Rightarrow \dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{{{x}^{4}}-1}=\dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}+1 \right)}
Now, divide the two fractions by taking the reciprocal of the second fraction,
⇒xx2+1÷x(x2−1)(x2+1)=xx2+1×(x2−1)(x2+1)x\Rightarrow \dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}+1 \right)}=\dfrac{x}{{{x}^{2}}+1}\times \dfrac{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}+1 \right)}{x}
Simplify the expression by cancelling all the like terms present in the numerator and denominator which is (x2+1)\left( {{x}^{2}}+1 \right) and xx to get,
⇒xx2+1×(x2−1)(x2+1)x=(x2−1)\Rightarrow \dfrac{x}{{{x}^{2}}+1}\times \dfrac{\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}+1 \right)}{x}=\left( {{x}^{2}}-1 \right)
Therefore, the given expression xx2+1÷xx4−1\dfrac{x}{{{x}^{2}}+1}\div \dfrac{x}{{{x}^{4}}-1}can be simplified to(x2−1)\left( {{x}^{2}}-1 \right).
The domain of a function can be stated as the range of all possible numbers where the given function will be defined and will have a value. While finding the domain of functions, equate the function to zero and find the values for which the function will not be defined.
For the function (x2−1)\left( {{x}^{2}}-1 \right), we can find the values where the function will not be defined by equation it to zero, therefore,
⇒(x2−1)=0\Rightarrow \left( {{x}^{2}}-1 \right)=0
By solving the above equation, we get the values of xx as,
⇒x2=1\Rightarrow {{x}^{2}}=1
Taking square roots on both sides,
⇒x=±1\Rightarrow x=\pm 1
Therefore, the given function will not be defined when x=±1,0x=\pm 1,0. Therefore, we get the value of the domain as (x∣x∈R,x≠±1,0)\left( x|x\in R,x\ne \pm 1,0 \right)

Note: While finding the domain of any function, always remember that if there is a fraction involved in the function then the denominator of the fraction cannot be zero. Also, if there is a value under the square root sign then it should have a positive value, otherwise the number will be considered as an imaginary number.