Multiple Choice
Graph the given function.
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6. Exponential & Logarithmic Functions
Graphing Logarithmic Functions
Multiple Choice
Which of the following statements correctly describes the graph of (where denotes base-10 logarithm)?
A
It is the graph of shifted left 1 unit, with domain and vertical asymptote .
B
It is the reflection of across the -axis, with domain and vertical asymptote .
C
It is the reflection of across the origin, with domain and vertical asymptote .
D
It is the reflection of across the -axis, with domain and vertical asymptote .
Verified step by step guidance1
Recall the basic graph of the logarithmic function \(y = \log(x)\), which has a domain of \(x > 0\) and a vertical asymptote at \(x = 0\).
Consider the function \(y = \log(-x)\). To understand its graph, analyze the inside of the logarithm: \(-x\) must be positive, so the domain is \(-x > 0\), which simplifies to \(x < 0\).
Since the input to the logarithm is \(-x\), this means the graph of \(y = \log(-x)\) is obtained by reflecting the graph of \(y = \log(x)\) across the \(y\)-axis (because replacing \(x\) by \(-x\) reflects points horizontally).
The vertical asymptote of \(y = \log(x)\) is at \(x = 0\). Reflecting across the \(y\)-axis does not change the location of the vertical asymptote, so \(y = \log(-x)\) also has a vertical asymptote at \(x = 0\).
Summarize: The graph of \(y = \log(-x)\) is the reflection of \(y = \log(x)\) across the \(y\)-axis, with domain \(x < 0\) and vertical asymptote at \(x = 0\).
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