Even and odd functions - Relations and Functions

Even and odd functions

If we are asked whether a given graph is symmetrical about the y-axis or not, it's easy to answer because we only need to see if there is a mirror image about the y-axis or not. But what if we are only given a function, but not the graph? In this section, we will broaden our knowledge about symmetry in functions while classifying symmetries algebraically, as well as learning the notion of odd and even functions.

Basic Concepts
Reflection across the y-axis: $y = f(-x)$
Reflection across the x-axis: $y = -f(x)$

Related Concepts
Sine graph: y = sin x
Cosine graph: y = cos x
Definite integral

Lessons

Notes:

When $f(-x) = f(x),$ function is even
$f(-x) = -f(x),$ function is odd

Even and odd functions

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AlgebraReflection across the y-axis: y=f(−x)y = f(-x)y=f(−x)

AlgebraReflection across the x-axis: y=−f(x)y = -f(x)y=−f(x)

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Reflection across the y-axis: y=f(−x)y = f(-x)y=f(−x)

Reflection across the x-axis: y=−f(x)y = -f(x)y=−f(x)

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Don't just watch, practice makes perfect

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Lessons

Notes:

When f(−x)=f(x),f(-x) = f(x), f(−x)=f(x), function is even f(−x)=−f(x),f(-x) = -f(x), f(−x)=−f(x), function is odd

Intro Lesson

What are even and odd functions? • How to determine if it is an even or odd function graphically and algebraically?

1.

Determine if the function f(x)=7x9+12f(x)=7x^9+12 f(x)=7x9+12 is even, odd, or neither

2.

Determine if the function f(x)=3x7+4x5−90x2f(x)=3x^7+4x^5-90x^2 f(x)=3x7+4x5−90x2 is even, odd, or neither

3.

Determine if the function f(x)=400xsin(x) f(x)=400xsin(x) f(x)=400xsin(x) is even, odd, or neither

4.

Determine if the function y=4x8+2x4−7x2y=4x^8+2x^4-7x^2 y=4x8+2x4−7x2 is even, odd, or neither

5.

Determine if the function y=7csc(x)+2tanxy=7csc(x)+2tanx y=7csc(x)+2tanx is even, odd, or neither

Even and odd functions

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Reflection across the y-axis: $y = f(-x)$

Algebra
Reflection across the x-axis: $y = -f(x)$

Reflection across the y-axis: $y = f(-x)$

Reflection across the x-axis: $y = -f(x)$

or

When $f(-x) = f(x),$ function is even
$f(-x) = -f(x),$ function is odd

What are even and odd functions?
• How to determine if it is an even or odd function graphically and algebraically?

Determine if the function $f(x)=7x^9+12$ is even, odd, or neither

Determine if the function $f(x)=3x^7+4x^5-90x^2$ is even, odd, or neither

Determine if the function $f(x)=400xsin(x)$ is even, odd, or neither

Determine if the function $y=4x^8+2x^4-7x^2$ is even, odd, or neither

Determine if the function $y=7csc(x)+2tanx$ is even, odd, or neither