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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 63
Capitolo 5, Problema 63

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.
Graph of f(x) = log x showing points (1,0), (5,0.7), (10,1) and vertical asymptote at x = 0.
g(x) = 1-log x

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Start with the graph of f(x) = \(\log\) x, which has a vertical asymptote at x = 0.
The function g(x) = 1 - \(\log\) x is a vertical reflection of f(x) = \(\log\) x across the x-axis, followed by a vertical shift upwards by 1 unit.
Reflect the graph of f(x) = \(\log\) x across the x-axis to get -\(\log\) x.
Shift the graph of -\(\log\) x upwards by 1 unit to obtain g(x) = 1 - \(\log\) x.
The domain of g(x) is (0, \(\infty\)) and the range is (-\(\infty\), 1). The vertical asymptote remains at x = 0.

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Logarithmic Function and Its Graph

The logarithmic function f(x) = log x is the inverse of the exponential function and is defined only for positive x-values. Its graph passes through (1,0) and has a vertical asymptote at x = 0. Understanding this base graph is essential for analyzing transformations and domain restrictions.
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Graphs of Logarithmic Functions

Transformations of Functions

Transformations include shifts, reflections, and stretches of the base graph. For g(x) = 1 - log x, the graph is reflected vertically (due to the negative sign) and shifted upward by 1 unit. Recognizing these changes helps in sketching the new graph and determining its properties.
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Domain & Range of Transformed Functions

Asymptotes, Domain, and Range

The vertical asymptote of log functions remains at x = 0 despite transformations. The domain is the set of x-values where the function is defined (x > 0), and the range depends on vertical shifts and reflections. Identifying asymptotes and domain/range is crucial for graphing and understanding function behavior.
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Determining Horizontal Asymptotes
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