Determine if Odd, Even, or Neither h(x)=3x^4-10x^3+x^2-x

$h (x) = 3 x^{4} - 10 x^{3} + x^{2} - x$

Step 1

Find $f (- x)$ .

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

$h (- x) = 3 {(- x)}^{4} - 10 {(- x)}^{3} + {(- x)}^{2} - (- x)$

Simplify each term.

Apply the product rule to $- x$ .

$h (- x) = 3 ({(- 1)}^{4} x^{4}) - 10 {(- x)}^{3} + {(- x)}^{2} - (- x)$

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

$h (- x) = 3 (1 x^{4}) - 10 {(- x)}^{3} + {(- x)}^{2} - (- x)$

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

$h (- x) = 3 x^{4} - 10 {(- x)}^{3} + {(- x)}^{2} - (- x)$

Apply the product rule to $- x$ .

$h (- x) = 3 x^{4} - 10 ({(- 1)}^{3} x^{3}) + {(- x)}^{2} - (- x)$

Raise $- 1$ to the power of $3$ .

$h (- x) = 3 x^{4} - 10 (- x^{3}) + {(- x)}^{2} - (- x)$

Multiply $- 1$ by $- 10$ .

$h (- x) = 3 x^{4} + 10 x^{3} + {(- x)}^{2} - (- x)$

Apply the product rule to $- x$ .

$h (- x) = 3 x^{4} + 10 x^{3} + {(- 1)}^{2} x^{2} - (- x)$

Raise $- 1$ to the power of $2$ .

$h (- x) = 3 x^{4} + 10 x^{3} + 1 x^{2} - (- x)$

Multiply $x^{2}$ by $1$ .

$h (- x) = 3 x^{4} + 10 x^{3} + x^{2} - (- x)$

Multiply $- (- x)$ .

Multiply $- 1$ by $- 1$ .

$h (- x) = 3 x^{4} + 10 x^{3} + x^{2} + 1 x$

Multiply $x$ by $1$ .

$h (- x) = 3 x^{4} + 10 x^{3} + x^{2} + x$

Step 2

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

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

$3 x^{4} + 10 x^{3} + x^{2} + x$ $\neq$ $3 x^{4} - 10 x^{3} + x^{2} - x$

Since $3 x^{4} + 10 x^{3} + x^{2} + x$ $\neq$ $3 x^{4} - 10 x^{3} + x^{2} - x$ , the function is not even.

The function is not even

Step 3

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

Find $- f (x)$ .

Multiply $3 x^{4} - 10 x^{3} + x^{2} - x$ by $- 1$ .

$- f (x) = - (3 x^{4} - 10 x^{3} + x^{2} - x)$

Apply the distributive property.

$- f (x) = - (3 x^{4}) - (- 10 x^{3}) - x^{2} + x$

Simplify.

Multiply $3$ by $- 1$ .

$- f (x) = - 3 x^{4} - (- 10 x^{3}) - x^{2} + x$

Multiply $- 10$ by $- 1$ .

$- f (x) = - 3 x^{4} + 10 x^{3} - x^{2} + x$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$- f (x) = - 3 x^{4} + 10 x^{3} - x^{2} + 1 x$

Multiply $x$ by $1$ .

$- f (x) = - 3 x^{4} + 10 x^{3} - x^{2} + x$

Since $3 x^{4} + 10 x^{3} + x^{2} + x$ $\neq$ $- 3 x^{4} + 10 x^{3} - x^{2} + x$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5

Determine if Odd, Even, or Neither h(x)=3x^4-10x^3+x^2-x

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