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# Symmetry

## Even and odd functions are symmetric across the y axis or about the origin.

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Practice Symmetry
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Symmetry

Some functions, like the sine function, the absolute value function and the squaring function, have reflection symmetry across the line x=0\begin{align*}x=0\end{align*}. Other functions like the cubing function and the reciprocal function have rotational symmetry about the origin.

Why is the first group categorized as even functions while the second group is categorized as odd functions?

#### Watch This

http://www.youtube.com/watch?v=8VgmBe3ulb8 Khan Academy: Recognizing Odd and Even Functions

#### Guidance

Functions symmetrical across the line x=0\begin{align*}x=0\end{align*} (the y\begin{align*}y\end{align*} axis) are called even. Even functions have the property that when a negative value is substituted for x\begin{align*}x\end{align*}, it produces the same value as when the positive value is substituted for the x\begin{align*}x\end{align*}.

f(x)=f(x)\begin{align*}f(-x)=f(x)\end{align*}

Functions that have rotational symmetry about the origin are called odd functions. When a negative x\begin{align*}x\end{align*} value is substituted into the function, it produces a negative version of the function evaluated at a positive value.

f(x)=f(x)\begin{align*}f(-x)=-f(x)\end{align*}

This property becomes increasingly important in problems and proofs of Calculus and beyond, but for now it is sufficient to identify functions that are even, odd or neither and show why.

Example A

Show that f(x)=3x45x2+1\begin{align*}f(x)=3x^4-5x^2+1\end{align*} is even.

Solution:

f(x)=3(x)45(x)2+1=3x45x2+1=f(x)

The property that both positive and negative numbers raised to an even power are always positive is the reason why the term even is used. It does not matter that the coefficients are even or odd, just the exponents.

Example B

Show that f(x)=4x3x\begin{align*}f(x)=4x^3-x\end{align*} is odd.

Solution:

f(x)=4(x)3x=4x3+x=(4x3+x)=f(x)

Just like even functions are named, odd functions are named because negative signs don’t disappear and can always be factored out of odd functions.

Example C

Identify whether the function is even, odd or neither and explain why.

f(x)=4x3|x|\begin{align*}f(x)=4x^3-|x|\end{align*}

Solution:

f(x)=4(x)3x=4x3x

This does not seem to match either f(x)=4x3|x|\begin{align*}f(x)=4x^3-|x|\end{align*} or f(x)=4x3+|x|\begin{align*}-f(x)=-4x^3+|x|\end{align*}. Therefore, this function is neither even nor odd.

Note that this function is a difference of an odd function and an even function. This should be a clue that the resulting function is neither even nor odd.

Concept Problem Revisited

Even and odd functions describe different types of symmetry, but both derive their name from the properties of exponents. A negative number raised to an even number will always be positive. A negative number raised to an odd number will always be negative.

#### Vocabulary

An even function means f(x)=f(x)\begin{align*}f(-x)=f(x)\end{align*}. Even functions have reflection symmetry across the line x=0\begin{align*}x=0\end{align*}

An odd function means f(x)=f(x)\begin{align*}f(-x)=-f(x)\end{align*}. Odd functions have rotation symmetry about the origin.

#### Guided Practice

1. Which of the basic functions are even, which are odd and which are neither?

2. Suppose h(x)\begin{align*}h(x)\end{align*} is an even function and g(x)\begin{align*}g(x)\end{align*} is an odd function. f(x)=h(x)+g(x).\begin{align*}f(x)=h(x)+g(x). \end{align*} Is f(x)\begin{align*}f(x)\end{align*} even or odd?

3. Determine whether the following function is even, odd, or neither.

f(x)=x(x21)(x4+1)\begin{align*}f(x)=x(x^2-1)(x^4+1)\end{align*}

Answers:

1. Even Functions: The squaring function, the absolute value function.

Odd Functions: The identity function, the cubing function, the reciprocal function, the sine function.

Neither: The square root function, the exponential function and the log function. The logistic function is also neither because it is rotationally symmetric about the point (0,12)\begin{align*}\left(0, \frac{1}{2}\right)\end{align*} as opposed to the origin.

2. If h(x)\begin{align*}h(x)\end{align*} is even then h(x)=h(x)\begin{align*}h(-x)=h(x)\end{align*}. If g(x)\begin{align*}g(x)\end{align*} is odd then g(x)=g(x)\begin{align*}g(-x)=-g(x)\end{align*}

Therefore: f(x)=h(x)+g(x)=h(x)g(x)\begin{align*}f(-x)=h(-x)+g(-x)=h(x)-g(x)\end{align*}

This does not match f(x)=h(x)+g(x)\begin{align*}f(x)= h(x)+g(x)\end{align*} nor does it match f(x)=h(x)g(x)\begin{align*}-f(x)=-h(x)-g(x)\end{align*}

This is a proof that shows the sum of an even function and an odd function will never itself be even or odd.

3.

f(x)f(x)=x(x21)(x4+1)=(x)((x)21)((x)4+1)=x(x21)(x4+1)=f(x)

The function is odd.

#### Practice

Determine whether the following functions are even, odd, or neither.

1. f(x)=4x2+1\begin{align*}f(x)=-4x^2+1\end{align*}

2. g(x)=5x33x\begin{align*}g(x)=5x^3-3x\end{align*}

3. h(x)=2x2x\begin{align*}h(x)=2x^2-x\end{align*}

4. j(x)=(x4)(x3)3\begin{align*}j(x)=(x-4) (x-3)^3\end{align*}

5. k(x)=x(x21)2\begin{align*}k(x)=x(x^2-1)^2\end{align*}

6. f(x)=2x35x22x+1\begin{align*}f(x)=2x^3-5x^2-2x+1\end{align*}

7. g(x)=2x24x+2\begin{align*}g(x)=2x^2-4x+2\end{align*}

8. h(x)=5x4+x2+2\begin{align*}h(x)=-5x^4+x^2+2\end{align*}

9. Suppose h(x)\begin{align*}h(x)\end{align*} is even and g(x)\begin{align*}g(x)\end{align*} is odd. Show that f(x)=h(x)g(x)\begin{align*}f(x)=h(x)-g(x)\end{align*} is neither even nor odd.

10. Suppose h(x)\begin{align*}h(x)\end{align*} is even and g(x)\begin{align*}g(x)\end{align*} is odd. Show that f(x)=h(x)g(x)\begin{align*}f(x)=\frac{h(x)}{g(x)}\end{align*} is odd.

11. Suppose h(x)\begin{align*}h(x)\end{align*} is even and g(x)\begin{align*}g(x)\end{align*} is odd. Show that f(x)=h(x)g(x)\begin{align*}f(x)=h(x) \cdot g(x)\end{align*} is odd.

12. Is the sum of two even functions always an even function? Explain.

13. Is the sum of two odd functions always an odd function? Explain.

14. Why are some functions neither even nor odd?

15. If you know that a function is even or odd, what does that tell you about the symmetry of the function?

### Vocabulary Language: English

Even Function

Even Function

An even function is a function with a graph that is symmetric with respect to the $y$-axis and has the property that $f(-x) = f(x)$.
reflection symmetry

reflection symmetry

A figure has reflection symmetry if it can be reflected across a line and look exactly the same as it did before the reflection.
Rotation Symmetry

Rotation Symmetry

A figure has rotational symmetry if it can be rotated less than $360^\circ$ around its center point and look exactly the same as it did before the rotation.

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