Determine if Odd, Even, or Neither -x^40+1

$- x^{40} + 1$

Step 1

Write $- x^{40} + 1$ as a function.

$f (x) = - x^{40} + 1$

Step 2

Find $f (- x)$ .

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

$f (- x) = - {(- x)}^{40} + 1$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - ({(- 1)}^{40} x^{40}) + 1$

Multiply $- 1$ by ${(- 1)}^{40}$ by adding the exponents.

Move ${(- 1)}^{40}$ .

$f (- x) = {(- 1)}^{40} \cdot (- 1 x^{40}) + 1$

Multiply ${(- 1)}^{40}$ by $- 1$ .

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

$f (- x) = {(- 1)}^{40} \cdot ((- 1) x^{40}) + 1$

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

$f (- x) = {(- 1)}^{40 + 1} x^{40} + 1$

Add $40$ and $1$ .

$f (- x) = {(- 1)}^{41} x^{40} + 1$

Raise $- 1$ to the power of $41$ .

$f (- x) = - x^{40} + 1$

Step 3

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

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

Since $- x^{40} + 1 = - x^{40} + 1$ , the function is even.

The function is even

Step 4