Question

Sketch the graph of $f$ $$f(x)=3^{x}-3^{-x}$$

   Sketch the graph of $f$
$$f(x)=3^{x}-3^{-x}$$

Precalculus: Functions and Graphs

Precalculus: Functions and Graphs

Earl W. Swokowski,… 11th Edition

Chapter 4, Problem 24

Instant Answer

Solved by Expert Harmender Singh Yadav

10/22/2020

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Step 1: First, we need to understand the behavior of the function $f(x)=3^{x}-3^{-x}$. Show more…

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Sketch the graph of $f$ $$f(x)=3^{x}-3^{-x}$$

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Key Concepts

Exponential Functions

Exponential functions have the form a^x, where a is a positive constant different from 1. They are characterized by rapid growth or decay depending on whether a is greater than or less than 1. Understanding their inherent properties—such as their domains, ranges, and long-term behavior—is essential when sketching graphs and analyzing more complex functions built from them.

Transformation of Exponential Functions

When exponential functions are combined through operations such as addition or subtraction, the resulting function embodies modified growth patterns and unique features. These transformations can alter the rate of increase or decrease, reposition key points on the graph, and affect asymptotic behavior, making it important to understand how each component function contributes to the overall shape.

Odd Functions and Symmetry

An odd function is defined by the property f(-x) = -f(x), which implies that its graph has rotational symmetry about the origin. Recognizing that a function is odd can simplify the sketching process, as the behavior in one half of the coordinate system directly informs the behavior in the other half.

Asymptotic Behavior

Evaluating the asymptotic behavior of a function involves understanding how it behaves as x approaches positive or negative infinity. This analysis helps identify whether the function approaches a horizontal line, diverges to infinity, or behaves in another predictable manner. Such insight is crucial for accurately sketching the overall graph and understanding its end behavior.

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