In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = (1/2)log₂ x

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Start by understanding the base function: \(f(x) = \log_{2} x\). This function has a vertical asymptote at \(x = 0\), a domain of \((0, \infty)\), and a range of \((-\infty, \infty)\).

Next, analyze the given function \(g(x) = \frac{1}{2} \log_{2} x\). Notice that this is a vertical scaling of the original function \(f(x)\) by a factor of \(\frac{1}{2}\).

To graph \(g(x)\), take the graph of \(f(x)\) and compress it vertically by multiplying all \(y\)-values by \(\frac{1}{2}\). This transformation does not affect the \(x\)-values or the vertical asymptote.

Since the vertical asymptote of \(f(x)\) is at \(x = 0\), and the transformation does not shift it horizontally, the vertical asymptote of \(g(x)\) remains at \(x = 0\).

Determine the domain and range of \(g(x)\). The domain remains \((0, \infty)\) because the logarithm is undefined for \(x \leq 0\). The range is all real numbers \((-\infty, \infty)\) because vertical scaling by \(\frac{1}{2}\) does not restrict the output values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Functions and Their Graphs

A logarithmic function, such as f(x) = log₂ x, is the inverse of an exponential function with base 2. Its graph passes through (1,0) and increases slowly, defined only for x > 0. Understanding its shape and key points is essential for graphing and analyzing transformations.

Transformations of Functions

Transformations include vertical stretches/compressions, reflections, and shifts applied to a base graph. For g(x) = (1/2)log₂ x, the factor 1/2 compresses the graph vertically, affecting the steepness but not the domain or vertical asymptote. Recognizing these changes helps in sketching the new graph accurately.

Vertical Asymptotes, Domain, and Range of Logarithmic Functions

Logarithmic functions have a vertical asymptote where the argument equals zero, here at x = 0. The domain is all positive real numbers (x > 0), and the range is all real numbers. Identifying the asymptote and these sets is crucial for understanding the function's behavior and graph.

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Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. (1/2)ln x - ln y

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Textbook Question

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)