Textbook Question
Give the equation of each exponential function whose graph is shown.
Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 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.
g(x) = 1-log x
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
5:26
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.
Recommended video:
4:22Domain & 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.
Recommended video:
4:48Determining Horizontal Asymptotes
Related Practice
Textbook Question
The figure shows the graph of f(x) = ln x. In Exercises 65–74, 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. g(x) = ln (x+2)
Textbook Question
In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64
Textbook Question
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.
Textbook Question
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. 6+2 ln x=5
Textbook Question
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. (1/2)(log x + log y)