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The domain of a function is the set of values that you can input into a function without breaking any mathematical rules. When working with logarithmic functions, it's crucial to ensure that the input is suitable for the function. For logarithmic functions like \( y = \log(-x) \), you need to make sure that the expression inside the logarithm is positive.
This ensures that you're not trying to take the logarithm of zero or a negative number, which is not possible in real numbers.
To find the domain, you'd solve any inequalities that arise from these restrictions. For example, in our function, we focus on \(-x > 0\), which means \(x\) must be less than zero.
This logical deduction lets us determine that the domain is all negative real numbers.

Inequalities are mathematical expressions that use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to show the relationship between two values or expressions. In the context of this function, you need to use inequalities to figure out which \(x\) values make the logarithmic function valid.
For \(y = \log(-x)\), the requirement is that \(-x > 0\). In other words, \(-x\) needs to be a positive number since the argument of a log function must be more than zero.
To solve this, we look at \(-x > 0\), which means multiplying both sides by \(-1\), reversing the inequality, leading to \(x < 0\).
This tells us that any number less than zero will work in this function, also known as negative numbers.

Interval notation provides a shorthand way to express a range of numbers in mathematics. When you're expressing the domain of a function, interval notation is handy. It's concise and conveys all the values that make the function work correctly.
For the logarithmic function \(y = \log(-x)\), the calculation led us to realize \(x < 0\). Expressing this in interval notation involves writing \((-fty, 0)\).
Here's a quick breakdown:

• \(-\infty\) implies starting from negative infinity, illustrating that \(x\) can be any small negative number.
• \(0\) means up to, but not including, zero.

With this interval notation, we efficiently capture all possible negative real numbers that can serve as inputs to the function.

Negative numbers are numbers less than zero. They're on the left side of zero when you think about a number line. In this function, \(y = \log(-x)\), the trick is to realize that the function is happy with inputs that are less than zero.
Why does this happen? It's because \(-x\) needs to be positive, so \(x\) must inherently be negative for \(-x\) to flip into a positive value.
Negative numbers are used in many real-world contexts, like indicating debt in finances or below-zero temperatures.
In math, they're crucial for cases like this: ensuring certain expressions correctly handle operations like logarithms, which require positive inputs.