/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Sketch a graph of the function a... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 77

Sketch a graph of the function and determine whether it is even, odd, or neither. Verify your answer algebraically. $$f(x)=-9$$

Short Answer

Expert verified
The function \(f(x)=-9\) is even, as illustrated by its graph and confirmed through algebraic verification.

Step by step solution

01

Sketch the Graph

To sketch the graph of the function \(f(x) = -9\), since the value of the function is not dependent on the input (x), it will be a straight horizontal line on the coordinate plane at y = -9.
02

Determine the Nature of the Function

To identify whether the function is even, odd, or neither, assess the function using the algebraic definitions: an even function is one for which \(f(x) = f(-x)\) for all x, while an odd function is one for which \(-f(x) = f(-x)\) for all x. Given \(f(x) = -9\) and \(f(-x) = -9\), it can be deduced that \(f(x) = f(-x)\). Therefore, the function is even.
03

Algebraic Verification

To confirm this algebraically, substitute x with -x in the original equation, that is, evaluate \(f(-x)\). If this results in the original function (i.e., \(f(-x) = f(x)\)), then the function is even. For \(f(x)=-9\), the substitution gives \(f(-x) = -9\), which is the original function. This confirms algebraically that the function is even.

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