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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 4 - Exponential and Logarithmic Functions문제 56
5장, 문제 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

검증된 단계별 안내
1
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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주요 개념

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

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.
추천 영상:
5:26
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.
추천 영상:
4:22
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.
추천 영상:
3:12
Determining Vertical Asymptotes
관련 실천
교과서 질문

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log3(x+4)=−3

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교과서 질문

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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교과서 질문

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 5 ln x - 2 ln y

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교과서 질문

Graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. f(x) = 2x, g(x) = 2-x

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교과서 질문

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log2(x+25)=4

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교과서 질문

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)=1+ log₂ x

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