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Chapter 3: Problem 44

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$y=\log (-x)$$

Short Answer

Expert verified
The domain of the function \(y=\log(-x) is \(x < 0\) (or \(-鈭� < x < 0\)). The x-intercept is at (-1,0) and the vertical asymptote is at \(x=0\).

Step by step solution

01

Identifying Domain of the Function

For the logarithmic function \(y=\log (-x)\), the argument -x must be greater than zero, as is the case with all logarithmic functions. If -x > 0, then x < 0. Therefore, the domain of the function is \(x < 0\), or more formally written as \((-鈭�, 0)\).
02

Finding x-intercept of the function

The x-intercept of a function is the point at which it crosses the x-axis. That means when \(y=0\). If we substitute \(y=0\) into the equation, we get \(0=\log(-x)\). The x for which this equation is true defines the x-intercept of the function. The only value that satisfies this equation is \(x=-1\), because the logarithm of 1, no matter the base, is always zero. Thus, the x-intercept of the function is at (-1,0).
03

Determining the vertical asymptote of the function

A vertical asymptote can be identified where the function is undefined. For \(y=\log(-x)\), the function is undefined when \(x=0\). Hence, the vertical asymptote of this function is \(x=0\).
04

Sketching the graph

Using all this information, a rough sketch of the graph can be drawn. On the domain of x < 0, we have a vertical asymptote at x=0 and an x-intercept at (-1,0) as its prominent features. The function gradually increases from -infinity to 0 as it moves from left to right.

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