Algebra Examples

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

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

Move $x^{2 - 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2.2

Multiply $x^{2}$ by $x^{- (2 - 4)}$ by adding the exponents.

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Step 2.2.1

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

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

Step 2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.3

Subtract $4$ from $2$ .

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

Step 2.2.4

Multiply $- 1$ by $- 2$ .

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

Step 2.2.5

Add $2$ and $2$ .

$f (x) = 4 x^{4}$

Step 3

Find $f (- x)$ .

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Step 3.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = 4 {(- x)}^{4}$

Step 3.2

Apply the product rule to $- x$ .

$f (- x) = 4 ({(- 1)}^{4} x^{4})$

Step 3.3

Raise $- 1$ to the power of $4$ .

$f (- x) = 4 (1 x^{4})$

Step 3.4

Multiply $x^{4}$ by $1$ .

$f (- x) = 4 x^{4}$

Step 4

A function is even if $f (- x) = f (x)$ .

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Step 4.1

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

Step 4.2

Since $4 x^{4} = 4 x^{4}$ , 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