Animated Solution for Mathematics - Functions: The domain of the function $f(x)=\frac{1}{\sqrt{|x|-x}}$ is

Welcome, fellow explorer of mathematics! Today, we are going to embark on a journey to uncover the domain of a function that might look intimidating at first glance:

Think of a function as a machine. You feed it an input, $x$, and it processes that input to give you an output. The domain is simply the set of all 'safe' inputs that we can feed into this machine without causing it to crash.

In mathematics, 'crashing' means encountering undefined operations, such as dividing by zero or taking the square root of a negative number.

To find the domain, we must identify the 'danger zones' in our function. Looking at the expression, we see two immediate constraints:

1. The Square Root Rule: The expression inside a square root must be non-negative. So, we must have $|x|-x\ge 0$.

2. The Denominator Rule: The denominator of a fraction cannot be zero. Since our square root is in the denominator, we must have $\sqrt{|x| - x} eq 0$, which implies $|x| - x eq 0$.

When we combine these two rules, the 'equal to' condition from the square root rule is cancelled out by the 'not equal to' condition from the denominator rule. This leaves us with one elegant, strict inequality:

This is our master condition.

A Tale of Two Cases

Now, how do we solve $|x|>x$? The absolute value function $|x|$ is a piecewise beast. It behaves differently depending on whether $x$ is positive or negative.

To tame it, we must break our analysis into two distinct cases.

The Non-negative Realm ($x\ge 0$)

Imagine we are in the world of positive numbers and zero. By definition, for any $x\ge 0$, $|x|=x$.

If we substitute this into our master inequality, we get:

Take a moment to think about this. Can any number be strictly greater than itself? Of course not! This is a mathematical contradiction.

Therefore, no value of $x\ge 0$ can ever satisfy our condition. The positive numbers and zero are officially rejected from our domain.

The Negative Realm ($x<0$)

Now, let us venture into the world of negative numbers. For any $x<0$, the definition of absolute value tells us that $|x|=-x$.

Let's substitute this into our master inequality:

Now, let's solve this algebraically. If we add $x$ to both sides, we get:

This result is beautiful! It perfectly matches the assumption we made for this case. Every single negative number satisfies the condition $|x|>x$.

We have systematically tested all real numbers. The positive numbers and zero led us to a contradiction, while the negative numbers welcomed us with open arms.

Thus, the function $f(x)$ is well-defined if and only if $x$ is strictly less than zero.

Our domain is the open interval $(-\infty ,0)$. You have successfully navigated the constraints, tamed the absolute value, and arrived at the solution.