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The graphs of f and g are given. Find a formula for the function g, if f(x) log2 x. g(x) :

The graphs of f and g are given. Find a formula for the function g, if f(x) log2 x. g(x) :

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
**Title: Understanding Logarithmic and Exponential Functions** **Description:** The problem involves analyzing the graphs of two functions, \( f \) and \( g \). We need to find a formula for the function \( g \) given that \( f(x) = \log_2 x \). **Graph Explanation:** - **Axes and Graph Elements:** - The graph consists of two curves labeled \( f \) and \( g \) plotted on the Cartesian plane. - The x-axis and y-axis are displayed with tick marks at intervals of 1. - **Function \( f(x) = \log_2 x \):** - The curve of \( f \) is shown in blue. This logarithmic function has a vertical asymptote at \( x = 0 \), depicted by a dashed blue line. - The graph of \( f \) starts from the lower left, passing through the origin, and curves upward to the right, indicating a decreasing slope as \( x \) increases. - **Function \( g(x) \):** - The curve of \( g \) is shown in red. This function is symmetric to function \( f \) along the line \( y = x \), suggesting that \( g(x) \) is the inverse of function \( f(x) \). - The graph of \( g \) starts from the lower right, passing through the origin, and curves upward to the left, indicating an increasing slope as \( x \) decreases. **Conclusion:** Since \( g(x) \) is the inverse of \( f(x) \) and given \( f(x) = \log_2 x \), we can determine that \( g(x) = 2^x \). This is because the inverse of the logarithmic base 2 function is the exponential base 2 function. **Key Concepts:** - The inverse of a logarithmic function \( \log_b x \) is the exponential function \( b^x \). - The properties of logarithmic and exponential functions include their respective behaviors and asymptotes on the graph. This detailed analysis of graphs enhances understanding of the relationship between logarithmic and exponential functions.
Transcribed Image Text:**Title: Understanding Logarithmic and Exponential Functions** **Description:** The problem involves analyzing the graphs of two functions, \( f \) and \( g \). We need to find a formula for the function \( g \) given that \( f(x) = \log_2 x \). **Graph Explanation:** - **Axes and Graph Elements:** - The graph consists of two curves labeled \( f \) and \( g \) plotted on the Cartesian plane. - The x-axis and y-axis are displayed with tick marks at intervals of 1. - **Function \( f(x) = \log_2 x \):** - The curve of \( f \) is shown in blue. This logarithmic function has a vertical asymptote at \( x = 0 \), depicted by a dashed blue line. - The graph of \( f \) starts from the lower left, passing through the origin, and curves upward to the right, indicating a decreasing slope as \( x \) increases. - **Function \( g(x) \):** - The curve of \( g \) is shown in red. This function is symmetric to function \( f \) along the line \( y = x \), suggesting that \( g(x) \) is the inverse of function \( f(x) \). - The graph of \( g \) starts from the lower right, passing through the origin, and curves upward to the left, indicating an increasing slope as \( x \) decreases. **Conclusion:** Since \( g(x) \) is the inverse of \( f(x) \) and given \( f(x) = \log_2 x \), we can determine that \( g(x) = 2^x \). This is because the inverse of the logarithmic base 2 function is the exponential base 2 function. **Key Concepts:** - The inverse of a logarithmic function \( \log_b x \) is the exponential function \( b^x \). - The properties of logarithmic and exponential functions include their respective behaviors and asymptotes on the graph. This detailed analysis of graphs enhances understanding of the relationship between logarithmic and exponential functions.
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