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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 57
Capitolo 5, Problema 57

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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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.
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Graphs of Logarithmic Functions

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.
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Domain & Range of Transformed Functions

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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Determining Vertical Asymptotes
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