Animated Solution for Mathematics - Functions: Domain of definition of the function $f(x)=\frac{3}{4-x^2}+{\log }_{10}(x^3-x)$, is

Defining the Mathematical Ecosystem

Welcome, aspiring engineer! Today, we are embarking on a journey to understand the 'domain' of a function. Think of a function not just as a cold, algebraic expression, but as a living, breathing ecosystem.

Every function has its own set of rules—its own laws of physics, if you will. The domain is simply the set of all possible inputs ($x$-values) for which this ecosystem can survive without collapsing into a singularity or producing an impossible result.

Our problem today features the function:

This function is a partnership between two very different entities: a rational fraction and a logarithmic term. For the function to exist, both partners must be satisfied. If one partner fails, the whole function fails. This is why we seek the intersection of their individual domains.

Avoiding the Singularity

Let us first examine the rational term: $\frac{3}{4-x^2}$. In the world of mathematics, division by zero is the ultimate taboo. It creates a singularity—a point where the function shoots off to infinity.

To keep our ecosystem stable, we must ensure the denominator is never zero:

Solving this is straightforward. We find that $x^2 eq 4$, which implies $x eq 2$ and $x eq -2$. These two points are our 'forbidden zones'. If $x$ ever touches these values, the rational part of our function will cease to exist.

The Realm of Positivity

Now, let us turn to the second partner: ${\log }_{10}(x^3-x)$. Logarithms are notoriously picky. They refuse to accept zero or negative numbers as inputs.

The argument of the logarithm must be strictly positive. Thus, we establish our second condition:

This is a cubic inequality. It might look intimidating, but let us break it down. We can factorize this expression to reveal its hidden structure:

Recognizing the difference of squares, we further refine this to:

This is the key to the entire problem. We have three critical points: $x=-1$, $x=0$, and $x=1$. These points are where the expression equals zero, acting as boundaries between regions of positive and negative values.

Visualizing the Flow

To solve this inequality, we use the Wavy Curve Method. Imagine the real number line divided into four segments by our critical points: $(-\infty ,-1)$, $(-1,0)$, $(0,1)$, and $(1,\infty )$.

1. For $x>1$: All three factors $(x)$, $(x-1)$, and $(x+1)$ are positive. Their product is positive. The curve is above the axis. 2. For $0<x<1$: The factor $(x-1)$ becomes negative, while the others remain positive. The product is negative. The curve dips below the axis. 3. For $-1<x<0$: The factors $(x-1)$ and $(x)$ are negative, while $(x+1)$ is positive. Two negatives make a positive! The product is positive. The curve rises above the axis. 4. For $x<-1$: All three factors are negative. The product of three negatives is negative. The curve is below the axis.

Since we require the expression to be strictly positive ($>0$), we select the regions where the curve is above the axis: $(-1,0)\cup (1,\infty )$.

Synthesizing the Truth

We are almost there! We have two conditions that must be satisfied simultaneously: 1. Rational Condition: $x\in \mathbb{R}\setminus \{-2,2\}$ 2. Logarithmic Condition: $x\in (-1,0)\cup (1,\infty )$

Now, we find the intersection. Look at the logarithmic domain: $(-1,0)\cup (1,\infty )$. Does it contain $-2$? No, $-2$ is far to the left of $-1$. So, the rational condition $x eq -2$ is already satisfied by the logarithmic condition.

However, look at the interval $(1,\infty )$. Does it contain $2$? Yes, it does! Since $x=2$ is forbidden by the rational part, we must remove it from this interval. This splits $(1,\infty )$ into two parts: $(1,2)$ and $(2,\infty )$.

Combining everything, our final domain is:

This is the complete, valid range of inputs for our function. You have successfully navigated the constraints, visualized the inequalities, and synthesized the final result. This is the essence of mathematical problem-solving—taking complex, disparate rules and weaving them into a single, elegant truth.