The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = log(x − 1)

검증된 단계별 안내

Start with the base function \(f(x) = \log x\), which has a vertical asymptote at \(x = 0\), domain \((0, \infty)\), and range \((-\infty, \infty)\).

Recognize that the function \(g(x) = \log(x - 1)\) is a horizontal shift of \(f(x)\) to the right by 1 unit because of the \((x - 1)\) inside the logarithm.

To find the new vertical asymptote, set the argument of the logarithm equal to zero: \(x - 1 = 0\), which gives \(x = 1\). This is the vertical asymptote of \(g(x)\).

Determine the domain of \(g(x)\) by finding where the argument of the logarithm is positive: \(x - 1 > 0\), so the domain is \((1, \infty)\).

Since logarithmic functions have a range of all real numbers, the range of \(g(x)\) remains \((-\infty, \infty)\).

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주요 개념

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Logarithmic Functions and Their Graphs

A logarithmic function, such as f(x) = log x, is the inverse of an exponential function. Its graph passes through (1,0) and has a vertical asymptote at x = 0. Understanding the basic shape and properties of log functions is essential for analyzing transformations.

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the graph of a function. For g(x) = log(x − 1), the graph of f(x) = log x shifts right by 1 unit. Recognizing how changes inside the function's argument affect the graph helps in sketching and identifying new asymptotes.

Domain, Range, and Asymptotes of Logarithmic Functions

The domain of log functions is restricted to positive inputs, so for g(x) = log(x − 1), the domain is x > 1. The range remains all real numbers. The vertical asymptote shifts accordingly, here to x = 1, marking where the function is undefined and the graph approaches infinity.