Precalculus Examples

f (x) = 2 x^{\frac{3}{7}}

$f (x) = 2 x^{\frac{3}{7}}$

Step 1

Find

f (- x)

$f (- x)$ .

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Step 1.1

Find

f (- x)

$f (- x)$ by substituting

- x

$- x$ for all occurrence of

x

$x$ in

f (x)

$f (x)$ .

f (- x) = 2 {(- x)}^{\frac{3}{7}}

$f (- x) = 2 {(- x)}^{\frac{3}{7}}$

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to

- x

$- x$ .

f (- x) = 2 ({(- 1)}^{\frac{3}{7}} x^{\frac{3}{7}})

$f (- x) = 2 ({(- 1)}^{\frac{3}{7}} x^{\frac{3}{7}})$

Step 1.2.2

Rewrite

- 1

$- 1$ as

{(- 1)}^{7}

${(- 1)}^{7}$ .

f (- x) = 2 ({({(- 1)}^{7})}^{\frac{3}{7}} x^{\frac{3}{7}})

$f (- x) = 2 ({({(- 1)}^{7})}^{\frac{3}{7}} x^{\frac{3}{7}})$

Step 1.2.3

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

f (- x) = 2 ({(- 1)}^{7 (\frac{3}{7})} x^{\frac{3}{7}})

$f (- x) = 2 ({(- 1)}^{7 (\frac{3}{7})} x^{\frac{3}{7}})$

f (- x) = 2 ({(- 1)}^{7 (\frac{3}{7})} x^{\frac{3}{7}})

$f (- x) = 2 ({(- 1)}^{7 (\frac{3}{7})} x^{\frac{3}{7}})$

Step 1.3

Cancel the common factor of

7

$7$ .

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Step 1.3.1

Cancel the common factor.

$f (- x) = 2 ({(- 1)}^{7 (\frac{3}{7})} x^{\frac{3}{7}})$

Step 1.3.2

Rewrite the expression.

$f (- x) = 2 ({(- 1)}^{3} x^{\frac{3}{7}})$

Step 1.4

Simplify the expression.

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Step 1.4.1

Raise

$- 1$ to the power of

$3$ .

$f (- x) = 2 (- x^{\frac{3}{7}})$

Step 1.4.2

Multiply

$- 1$ by

$2$ .

$f (- x) = - 2 x^{\frac{3}{7}}$

Step 2

A function is even if

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

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Step 2.1

Check if

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

Step 2.2

Since

$- 2 x^{\frac{3}{7}}$

$\neq$

$2 x^{\frac{3}{7}}$ , the function is not even.

The function is not even

Step 3

A function is odd if

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

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Step 3.1

Multiply

$2$ by

$- 1$ .

$- f (x) = - 2 x^{\frac{3}{7}}$

Step 3.2

Since

$- 2 x^{\frac{3}{7}} = - 2 x^{\frac{3}{7}}$ , the function is odd.

The function is odd

Step 4