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Understanding the domain of a function is crucial because it lets us know which values of \( x \) can be used in the function without resulting in an error. In the case of logarithmic functions like \( y = \log(-x) \), the rules of logarithms dictate that you cannot take the logarithm of zero or any negative number.

However, since the function here is \( \log(-x) \), you're actually looking for values of \( x \) that make \( -x \), the argument inside the log, positive. This means that \( x \) needs to be negative because multiplying a negative by another negative creates a positive.

Therefore, the domain for \( y = \log(-x) \) is all negative real numbers, or simply \( x < 0 \). Always remember:

• The argument of the log function must be positive.
• Identify transformations like "\(-x\)" to set the domain correctly.

A vertical asymptote in a logarithmic function is a line that the graph approaches but fundamentally never touches. It indicates where the function's value becomes infinitely large, either positively or negatively.

For the function \( y = \log(-x) \), the vertical asymptote occurs where the inside of the log function, or \(-x\), equals zero. Solving for \( x \) gives us \( x = 0 \). This means that at \( x = 0 \), the function is undefined because logarithms cannot process zero, and the graph will twirl toward infinity.

Key points to note:

• Vertical asymptotes occur where the log's argument would equal zero.
• It acts as a boundary; you can't include the asymptote value in the function's domain.

The \( x \)-intercept is the point where the graph of a function crosses the \( x \)-axis. Finding the \( x \)-intercept involves setting \( y = 0 \) and solving for \( x \).

In the function \( y = \log(-x) \), setting \( y = 0 \) gives us:\[0 = \log(-x)\]This logarithmic equation can be converted into its exponential form. Using the fact that the base of a common logarithm is 10, the equation becomes:\[-x = 10^0 = 1\]Thus, \( x = -1 \). Hence, the \( x \)-intercept occurs at the point \(( -1, 0 )\). This is where the graph touches the \( x \)-axis.

• Set \( y = 0 \) to find \( x \)-intercepts in general.
• Use exponential form conversion for logs: \( \log_b(x) = y \) turns into \( x = b^y \).

Graphing a function like \( y = \log(-x) \) involves visualizing how the function behaves over its domain and near its asymptotes. Here's a simple guide:

• Start by drawing the vertical asymptote at \( x = 0 \). This is your line of limitation.
• Place the \( x \)-intercept at \( x = -1 \), \( y = 0 \). Mark this point clearly on the graph.
• Sketch the curve, beginning from the \( x \)-intercept. It will swoop upwards, gently curving as \( x \) becomes more negative.
• The function will approach the vertical asymptote but shall never touch or cross it, always climbing toward \( y = \infty \).

Remember that the curve echoes the properties of logarithmic functions, growing very slowly and never actually reaching \( x = 0 \). Utilize graphing software or tools for a clearer visualization if you're unsure.