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Find the domain and range of each of the following functions  f(x)=\frac{x^{2}+x+1}{x^{2}+4x+3}

  • Option 1)

    y\in \left [ -\infty ,\frac{-2-\sqrt{7}}{2} \right ]\cup \left [ \frac{-2+\sqrt{7}}{2} ,\infty \right ],

     

     

     

  • Option 2)

    y\in (-10,10)

  • Option 3)

    y\in (-8,8)

  • Option 4)

    none of these 

 

Answers (1)

best_answer

As we learned

 

Rational Function -

f(x )= \frac{\rho \left ( x \right )}{q\left ( x \right )} Where \rho \left ( x \right )\: \: and\: \: q\left ( x \right ) polynomials in x.

 

 

- wherein

Domin of this function is R-\left \{ x:q\left ( x \right )= 0 \right \} Range depends on function.

 

 Here 

f(x)=\frac{x^{2}+x+1}{x^{2}+4x+3}   and f(x) is defined for all x\equiv R other numbers where x^{2}+4x+3=0

\Rightarrow x=-3,-1.

Hence the domain of f(x) =R-(-3,-1).

Let \frac{x^{2}+x+1}{x^{2}+4x+3}=y\Rightarrow x^{2}(1-y)+x(1-4y)+1-3y=0.

Since x is real, (1-4y)^{2}-4(1-y)(1-3y)\geq 0\Rightarrow 4y^{2}+8y-3\geq 0

\Rightarrow y\in \left [ -\infty ,\frac{-2-\sqrt{7}}{2} \right ]\cup \left [ \frac{-2+\sqrt{7}}{2},\infty \right ],  which is the required range

 


Option 1)

y\in \left [ -\infty ,\frac{-2-\sqrt{7}}{2} \right ]\cup \left [ \frac{-2+\sqrt{7}}{2} ,\infty \right ],

 

 

 

Option 2)

y\in (-10,10)

Option 3)

y\in (-8,8)

Option 4)

none of these 

Posted by

gaurav

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