Calculus Examples

f (x) = \sqrt{x^{4} - 16 x^{2}}

$f (x) = \sqrt{x^{4} - 16 x^{2}}$

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Simplify.

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Step 2.1

Factor

x^{2}

$x^{2}$ out of

x^{4} - 16 x^{2}

$x^{4} - 16 x^{2}$ .

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Step 2.1.1

Factor

x^{2}

$x^{2}$ out of

x^{4}

$x^{4}$ .

f (x) = \sqrt{x^{2} x^{2} - 16 x^{2}}

$f (x) = \sqrt{x^{2} x^{2} - 16 x^{2}}$

Step 2.1.2

Factor

x^{2}

$x^{2}$ out of

- 16 x^{2}

$- 16 x^{2}$ .

f (x) = \sqrt{x^{2} x^{2} + x^{2} \cdot - 16}

$f (x) = \sqrt{x^{2} x^{2} + x^{2} \cdot - 16}$

Step 2.1.3

Factor

x^{2}

$x^{2}$ out of

x^{2} x^{2} + x^{2} \cdot - 16

$x^{2} x^{2} + x^{2} \cdot - 16$ .

f (x) = \sqrt{x^{2} (x^{2} - 16)}

$f (x) = \sqrt{x^{2} (x^{2} - 16)}$

f (x) = \sqrt{x^{2} (x^{2} - 16)}

$f (x) = \sqrt{x^{2} (x^{2} - 16)}$

Step 2.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

f (x) = \sqrt{x^{2} (x^{2} - 4^{2})}

$f (x) = \sqrt{x^{2} (x^{2} - 4^{2})}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 4

$b = 4$ .

f (x) = \sqrt{x^{2} (x + 4) (x - 4)}

$f (x) = \sqrt{x^{2} (x + 4) (x - 4)}$

Step 2.4

Rewrite

x^{2} (x + 4) (x - 4)

$x^{2} (x + 4) (x - 4)$ as

x^{2} ((x + 2^{2}) (x - 4))

$x^{2} ((x + 2^{2}) (x - 4))$ .

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Step 2.4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

f (x) = \sqrt{x^{2} (x + 2^{2}) (x - 4)}

$f (x) = \sqrt{x^{2} (x + 2^{2}) (x - 4)}$

Step 2.4.2

Add parentheses.

f (x) = \sqrt{x^{2} ((x + 2^{2}) (x - 4))}

$f (x) = \sqrt{x^{2} ((x + 2^{2}) (x - 4))}$

f (x) = \sqrt{x^{2} ((x + 2^{2}) (x - 4))}

$f (x) = \sqrt{x^{2} ((x + 2^{2}) (x - 4))}$

Step 2.5

Pull terms out from under the radical.

f (x) = x \sqrt{(x + 2^{2}) (x - 4)}

$f (x) = x \sqrt{(x + 2^{2}) (x - 4)}$

Step 2.6

Raise

2

$2$ to the power of

2

$2$ .

f (x) = x \sqrt{(x + 4) (x - 4)}

$f (x) = x \sqrt{(x + 4) (x - 4)}$

f (x) = x \sqrt{(x + 4) (x - 4)}

$f (x) = x \sqrt{(x + 4) (x - 4)}$

Step 3

Find

f (- x)

$f (- x)$ .

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Step 3.1

Find

f (- x)

$f (- x)$ by substituting

- x

$- x$ for all occurrence of

x

$x$ in

f (x)

$f (x)$ .

f (- x) = (- x) \sqrt{((- x) + 4) ((- x) - 4)}

$f (- x) = (- x) \sqrt{((- x) + 4) ((- x) - 4)}$

Step 3.2

Remove parentheses.

f (- x) = - x \sqrt{(- x + 4) (- x - 4)}

$f (- x) = - x \sqrt{(- x + 4) (- x - 4)}$

f (- x) = - x \sqrt{(- x + 4) (- x - 4)}

$f (- x) = - x \sqrt{(- x + 4) (- x - 4)}$

Step 4

A function is even if

f (- x) = f (x)

$f (- x) = f (x)$ .

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Step 4.1

Check if

f (- x) = f (x)

$f (- x) = f (x)$ .

Step 4.2

Since

- x \sqrt{(- x + 4) (- x - 4)} = x \sqrt{(x + 4) (x - 4)}

$- x \sqrt{(- x + 4) (- x - 4)} = x \sqrt{(x + 4) (x - 4)}$ , the function is even.

The function is even

Step 5

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Step 6

Since the function is even, it is symmetric about the y-axis.

Y-axis symmetry

Step 7