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Graph the function f(x) = 2x-3. State the domain, range, and horizontal asymptote:

Graph the function f(x) = 2x-3. State the domain, range, and horizontal asymptote:

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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**Graphing Exponential Functions: A Practical Exercise** **Objective:** Graph the function \( f(x) = 2^{-x} - 3 \). State the domain, range, and horizontal asymptote. **Instructions:** 1. **Understand the Function:** The given function \( f(x) = 2^{-x} - 3 \) is an exponential function. Here, the base of the exponential function is 2, and it is raised to the power of \(-x\), shifted downward by 3 units. 2. **Graphing the Function:** - Create a table of values for \( x \) and corresponding \( f(x) \). - Plot these points on a graph. - Draw a smooth curve through these points to form the graph of the function. 3. **Domain and Range:** - **Domain:** The set of all possible input values (x-values) for the function. - **Range:** The set of all possible output values (y-values) for the function. 4. **Horizontal Asymptote:** A line that the graph of the function approaches but never touches as \( x \) approaches positive or negative infinity. **Detailed Steps:** - **Create a Table:** Choose a set of x-values (e.g., -2, -1, 0, 1, 2) and compute \( f(x) \) for each. - For \( x = -2 \): \( f(-2) = 2^{-(-2)} - 3 = 2^2 - 3 = 4 - 3 = 1 \) - For \( x = -1 \): \( f(-1) = 2^{-(-1)} - 3 = 2^1 - 3 = 2 - 3 = -1 \) - For \( x = 0 \): \( f(0) = 2^{-0} - 3 = 1 - 3 = -2 \) - For \( x = 1 \): \( f(1) = 2^{-1} - 3 = \frac{1}{2} - 3 = -2.5 \) - For \( x = 2 \): \( f(2) = 2^{-2} - 3 = \frac{1}{4} - 3 = -2.75 \
Transcribed Image Text:**Graphing Exponential Functions: A Practical Exercise** **Objective:** Graph the function \( f(x) = 2^{-x} - 3 \). State the domain, range, and horizontal asymptote. **Instructions:** 1. **Understand the Function:** The given function \( f(x) = 2^{-x} - 3 \) is an exponential function. Here, the base of the exponential function is 2, and it is raised to the power of \(-x\), shifted downward by 3 units. 2. **Graphing the Function:** - Create a table of values for \( x \) and corresponding \( f(x) \). - Plot these points on a graph. - Draw a smooth curve through these points to form the graph of the function. 3. **Domain and Range:** - **Domain:** The set of all possible input values (x-values) for the function. - **Range:** The set of all possible output values (y-values) for the function. 4. **Horizontal Asymptote:** A line that the graph of the function approaches but never touches as \( x \) approaches positive or negative infinity. **Detailed Steps:** - **Create a Table:** Choose a set of x-values (e.g., -2, -1, 0, 1, 2) and compute \( f(x) \) for each. - For \( x = -2 \): \( f(-2) = 2^{-(-2)} - 3 = 2^2 - 3 = 4 - 3 = 1 \) - For \( x = -1 \): \( f(-1) = 2^{-(-1)} - 3 = 2^1 - 3 = 2 - 3 = -1 \) - For \( x = 0 \): \( f(0) = 2^{-0} - 3 = 1 - 3 = -2 \) - For \( x = 1 \): \( f(1) = 2^{-1} - 3 = \frac{1}{2} - 3 = -2.5 \) - For \( x = 2 \): \( f(2) = 2^{-2} - 3 = \frac{1}{4} - 3 = -2.75 \
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