Skip to main content
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 40
5장, 문제 40

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)
Graph of f(x) = ln x (red) and g(x) = -ln(2x) (blue) with asymptotes and key points.
Graph of g(x) = -ln(2x) (red) and f(x) = ln x (blue) with transformations and key points.
Graph of f(x) = ln x (red) and g(x) = -ln(2x) (blue) with asymptotes and key points.
Graph of g(x) = -ln(2x) (red) and f(x) = ln x (blue) with transformations and key points.

검증된 단계별 안내
1
Step 1: Begin by analyzing the given functions f(x) = ln(x) and g(x) = -ln(2x). The function f(x) = ln(x) is the natural logarithm function, which has a vertical asymptote at x = 0 and is defined for x > 0. The function g(x) = -ln(2x) involves a reflection across the x-axis and a horizontal compression by a factor of 2.
Step 2: Identify the transformations applied to f(x) to obtain g(x). The negative sign in g(x) = -ln(2x) reflects the graph of f(x) across the x-axis, and the factor of 2 inside the logarithm compresses the graph horizontally. This means that for g(x), the x-values are scaled by a factor of 1/2 compared to f(x).
Step 3: Determine the asymptotes for both functions. For f(x) = ln(x), the vertical asymptote is at x = 0 because the natural logarithm is undefined for x ≤ 0. Similarly, g(x) = -ln(2x) also has a vertical asymptote at x = 0, as the argument of the logarithm (2x) must be positive.
Step 4: Use the graphs to determine the domain and range of each function. For f(x) = ln(x), the domain is (0, ∞) and the range is (-∞, ∞). For g(x) = -ln(2x), the domain is also (0, ∞), but the range is (-∞, ∞) due to the reflection across the x-axis.
Step 5: Plot key points to verify the transformations. For f(x), key points include (1, 0) and (e, 1). For g(x), the corresponding points after transformation are (0.5, 0) and (e/2, -1). These points confirm the horizontal compression and reflection applied to f(x) to obtain g(x).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
15m
도움이 되었나요?

주요 개념

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

Logarithmic Functions

Logarithmic functions, such as f(x) = ln(x), are the inverses of exponential functions. They are defined for positive real numbers and have a vertical asymptote at x = 0. Understanding their properties, including their domain, range, and behavior as x approaches the asymptote, is crucial for graphing and analyzing transformations.
추천 영상:
5:26
Graphs of Logarithmic Functions

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, g(x) = -ln(2x) represents a vertical reflection of f(x) = ln(x) and a horizontal compression. Recognizing how these transformations affect the graph helps in visualizing and deriving the new function's characteristics from the original.
추천 영상:
4:22
Domain & Range of Transformed Functions

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For logarithmic functions, vertical asymptotes occur where the function is undefined, such as at x = 0 for f(x) = ln(x). Identifying asymptotes is essential for understanding the behavior of the function at its boundaries and for determining the domain and range of the functions involved.
추천 영상:
6:24
Introduction to Asymptotes
관련 실천
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. log 5 + log 2

1032
views
교과서 질문

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = ex-1+2

781
views
교과서 질문

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(10x21−x37(x+1)2)\(\log\) \(\left\)( \(\frac{10x^2 \sqrt[3]{1 - x}\)}{7(x + 1)^2} \(\right\))

917
views
교과서 질문

Evaluate each expression without using a calculator. log5 57

806
views
교과서 질문

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e-x

1039
views
교과서 질문

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5(2x+3)=3(x−1)

898
views