Question

The domain of the function $f\left( x \right)=\dfrac{\left( \sqrt{-\log _{0.3}^{\left( x-1 \right)}} \right)}{\left( \sqrt{-{{x}^{2}}+2x+8} \right)}$
1.$\left( 1,4 \right)$
2.$\left( -2,4 \right)$
3.$\left[ 2,\left. 4 \right) \right.$
4. None of these

Answer 1

In order to find the domain of the given function $f\left( x \right)=\dfrac{\left( \sqrt{-\log _{0.3}^{\left( x-1 \right)}} \right)}{\left( \sqrt{-{{x}^{2}}+2x+8} \right)}$, firstly, we will be finding the roots of the given quadratic equation. Then we will be checking for the function at which they are defined for $f$. Then upon checking for them, we will be obtaining the domain of the function.

Complete step-by-step solution:
Now let us learn about the domain and range of functions. A domain of a function $f\left( x \right)$is the set of all values for which the function is defined and the range of the function is the set of all values that $f$ takes. We can also define the special functions whose domains are limited. The natural domain of a function is the set of all allowable input values. Natural domain is basically the x values for which the function is defined. 'Domain' or 'restricted domain' is 'man-made' .
Now let us find the domain of the given function i.e. $f\left( x \right)=\dfrac{\left( \sqrt{-\log _{0.3}^{\left( x-1 \right)}} \right)}{\left( \sqrt{-{{x}^{2}}+2x+8} \right)}$
Now let us find the roots of the equation $\left( \sqrt{-{{x}^{2}}+2x+8} \right)$

& -{{x}^{2}}+2x+8=0 \\\ & \Rightarrow {{x}^{2}}-2x-8=0 \\\ & \Rightarrow {{x}^{2}}+2x-4x-8=0 \\\ & \Rightarrow x\left( x+2 \right)-4\left( x+2 \right)=0 \\\ & \Rightarrow \left( x+2 \right)\left( x-4 \right) \\\ \end{aligned}$$ $$\therefore $$ The roots are $$\left( x+2 \right),\left( x-4 \right)$$. Now let us start checking for the domain. $$-\log _{0.3}^{\left( x-1 \right)}\ge 0,\left( x+2 \right)\left( x-4 \right)<0$$ Upon solving, $$\left( x-1 \right)>0$$ and $$x\ne -2,4$$ $$\begin{aligned} & \Rightarrow x\ge 2,-2 < x < 4,x >1 ,x\ne -2,4 \\\ & \Rightarrow x\in \left[ 2,\left. 4 \right) \right. \\\ \end{aligned}$$ $$\therefore $$ The domain of the function is $$\left[ 2,\left. 4 \right) \right.$$. **Hence option 3 is correct.** **Note:** We must note that for quadratic equations, we must be finding the roots in order to find the domain of any function. While finding the domain, the denominator of a fraction cannot be zero. While solving for a domain, the function must be checked for all possible values.