Animated Solution for Mathematics - Functions: The domain of the function $f(x)={\sin }^{-1}[2x^2-3]+{\log }_2\left({\log }_{\frac{1}{2}}(x^2-5x+5)\right)$, where $[t]$ is the greatest integer function, is :

To find the domain of the function $f(x)={\sin }^{-1}[2x^2-3]+{\log }_2({\log }_{\frac{1}{2}}(x^2-5x+5))$, we must satisfy every mathematical constraint simultaneously. We treat each component as a gatekeeper that restricts the possible values of $x$.

The Inverse Sine Gatekeeper

The domain of ${\sin }^{-1}(u)$ is restricted to the interval $[-1,1]$. Therefore, the argument must satisfy:

For the Greatest Integer Function (GIF) to fall within this range, the inner expression must satisfy:

Adding $3$ to all parts yields $2\le 2x^2<5$. Dividing by $2$, we obtain $1\le x^2<\frac{5}{2}$. Assuming $x>0$, we identify our first valid interval:

The Logarithmic Duality

Next, we address the outer logarithm: ${\log }_{\frac{1}{2}}(x^2-5x+5)>0$. Because the base $\frac{1}{2}$ is less than $1$, the inequality sign reverses when we remove the logarithm:

Rearranging this gives $x^2-5x+4<0$. Factoring the quadratic results in $(x-1)(x-4)<0$, which provides the interval:

We must also ensure the argument of the inner logarithm is strictly positive: $x^2-5x+5>0$. Solving $x^2-5x+5=0$ via the quadratic formula gives roots at $x=\frac{5\pm \sqrt{5}}{2}$. The quadratic is positive outside these roots:

The Final Intersection

To find the domain, we calculate the intersection $P_1\cap P_2\cap P_3$. First, we intersect $P_1=[1,\sqrt{\frac{5}{2}})$ and $P_3=(1,4)$ to get $(1,\sqrt{\frac{5}{2}})$.

Now, we intersect this result with $P_2$. Note that $\frac{5-\sqrt{5}}{2}\approx 1.38$ and $\sqrt{\frac{5}{2}}\approx 1.58$. Since $1<1.38<1.58$, the intersection of $(1,1.58)$ and $(-\infty ,1.38)$ is $(1,1.38)$.

The final domain is:

$$x\in \left(1,\frac{5-\sqrt{5}}{2}\right)$$