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. h(x) = log x − 1

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Start with the base function given: \(f(x) = \log x\). Recall that the graph of \(\log x\) has a vertical asymptote at \(x = 0\), a domain of \((0, \infty)\), and a range of \((-\infty, \infty)\).

Identify the transformation in the function \(h(x) = \log x - 1\). This represents a vertical shift downward by 1 unit of the original graph \(f(x) = \log x\).

Apply the vertical shift to the graph: move every point on the graph of \(f(x) = \log x\) down by 1 unit. This means the shape of the graph remains the same, but the entire curve is lowered by 1.

Determine the new equation of the asymptote. Since vertical shifts do not affect vertical asymptotes, the vertical asymptote remains at \(x = 0\).

Find the domain and range of \(h(x)\). The domain remains \((0, \infty)\) because the logarithm's input hasn't changed. The range shifts down by 1, so the new range is \((-\infty, \infty)\) shifted down by 1, which is still \((-\infty, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Logarithmic Functions

A logarithmic function, such as f(x) = log x, is the inverse of an exponential function. It is defined only for positive x-values and has a vertical asymptote at x = 0. Understanding its basic shape and properties is essential for graphing transformations.

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the graph of a function. For h(x) = log x − 1, subtracting 1 shifts the graph downward by 1 unit, affecting the range but not the domain or vertical asymptote.

Asymptotes, Domain, and Range

The vertical asymptote of log functions occurs where the argument is zero (x=0). The domain is the set of x-values where the function is defined (x > 0), and the range is all real numbers, which may shift with transformations.