Precalculus Examples

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

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Apply the product rule to $- x$ .

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

Step 1.2.2

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

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

Step 1.2.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.3

Cancel the common factor of $7$ .

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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