Question

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed. If $f(-x)=f(x)$ for all $x$ in the domain, we say that $f$is an _ function and symmetric to the axis. If $f(-x)=-f(x),$ the function is and symmetric to the .

Name: Problem 2, Chapter 2: Precalculus
Uploaded: 2020-05-20T18:29:25-08:00
Duration: 1 min 20 s
Channel: Dale Sanford
Description: Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed. If f(-x)=f(x) for all x in the domain, we say that fis an function and symmetric to the axis. If f(-x)=-f(x), the function is and symmetric to the .

   Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
If $f(-x)=f(x)$ for all $x$ in the domain, we say that $f$is an _____ function and symmetric to the ____ axis. If $f(-x)=-f(x),$ the function is ____ and symmetric to the ____.

Precalculus

John W. Coburn 2nd Edition

Chapter 2, Problem 2

Instant Answer

Solved by Expert Dale Sanford

05/20/2020

Step 1

If $f(-x)=f(x)$ for all $x$ in the domain, this means that the function $f$ is symmetric with respect to the y-axis. This is because the negative version of $x$ will be at the exact same height as the positive version. This is the definition of an even function. Show more…

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Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed. If $f(-x)=f(x)$ for all $x$ in the domain, we say that $f$is an _____ function and symmetric to the ____ axis. If $f(-x)=-f(x),$ the function is ____ and symmetric to the ____.

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Key Concepts

Even Function

An even function is one that satisfies the condition f(-x) = f(x) for all x in its domain. This means that the function's graph remains unchanged when reflected across the y-axis, indicating symmetry about that axis. It is a key concept in understanding function behavior and graph transformations.

Odd Function

An odd function is defined by the condition f(-x) = -f(x) for every x in its domain. This implies that the function's graph is symmetric with respect to the origin, meaning a rotation of 180 degrees about the origin will leave the graph unchanged.

Graph Symmetry

Graph symmetry in functions refers to the invariance of the graph under certain transformations, such as reflection or rotation. In the context of even and odd functions, even functions exhibit y-axis symmetry (reflection), while odd functions show origin symmetry (rotation), which aids in understanding the visual behavior and properties of functions.

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