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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 56
Capitolo 5, Problema 56

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. h(x) = 2 + log2x

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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 \(h(x) = 2 + \log_{2} x\). Notice that this is a vertical shift of the base function \(f(x)\) upward by 2 units.
Since adding 2 shifts the graph vertically, the vertical asymptote remains unchanged at \(x = 0\) because vertical asymptotes depend on the input values where the function is undefined, which is not affected by vertical shifts.
Determine the domain of \(h(x)\) by considering where \(\log_{2} x\) is defined. Since \(\log_{2} x\) is defined for \(x > 0\), the domain of \(h(x)\) is also \((0, \infty)\).
Determine the range of \(h(x)\) by shifting the range of \(f(x)\) up by 2. Since the range of \(f(x)\) is \((-\infty, \infty)\), the range of \(h(x)\) remains \((-\infty, \infty)\) because adding a constant shifts the entire range but does not restrict it.

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Graphing Logarithmic Functions

Graphing logarithmic functions involves plotting points based on the logarithm's definition and shape. For f(x) = log₂ x, the graph passes through (1,0), increases slowly, and has a vertical asymptote at x = 0. Understanding this base graph is essential before applying transformations.
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Graphs of Logarithmic Functions

Transformations of Functions

Transformations modify the graph of a base function by shifting, stretching, or reflecting it. For h(x) = 2 + log₂ x, the '+2' shifts the graph vertically upward by 2 units, affecting the range but not the vertical asymptote or domain.
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Domain & Range of Transformed Functions

Domain, Range, and Vertical Asymptotes of Logarithmic Functions

The domain of log₂ x is (0, ∞) because logarithms are undefined for non-positive values. The vertical asymptote is the line x = 0, where the function approaches negative infinity. Adding a constant shifts the range but does not change the domain or asymptote.
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Determining Vertical Asymptotes
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