Algebra Examples

$f (x) = {(x - 1)}^{2}$

Step 1

Find $f (- x)$ .

Tap for more steps...

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

$f (- x) = {((- x) - 1)}^{2}$

Rewrite ${(- x - 1)}^{2}$ as $(- x - 1) (- x - 1)$ .

$f (- x) = (- x - 1) (- x - 1)$

Expand $(- x - 1) (- x - 1)$ using the FOIL Method.

Tap for more steps...

Apply the distributive property.

$f (- x) = - x (- x - 1) - 1 (- x - 1)$

Apply the distributive property.

$f (- x) = - x (- x) - x \cdot - 1 - 1 (- x - 1)$

Apply the distributive property.

$f (- x) = - x (- x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Simplify and combine like terms.

Tap for more steps...

Simplify each term.

Tap for more steps...

Rewrite using the commutative property of multiplication.

$f (- x) = - 1 \cdot (- 1 x \cdot x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Move $x$ .

$f (- x) = - 1 \cdot (- 1 (x \cdot x)) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $x$ by $x$ .

$f (- x) = - 1 \cdot (- 1 x^{2}) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

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

$f (- x) = x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Multiply $- x \cdot - 1$ .

Tap for more steps...

Multiply $- 1$ by $- 1$ .

$f (- x) = x^{2} + 1 x - 1 (- x) - 1 \cdot - 1$

Multiply $x$ by $1$ .

$f (- x) = x^{2} + x - 1 (- x) - 1 \cdot - 1$

Multiply $- 1 (- x)$ .

Tap for more steps...

Multiply $- 1$ by $- 1$ .

$f (- x) = x^{2} + x + 1 x - 1 \cdot - 1$

Multiply $x$ by $1$ .

$f (- x) = x^{2} + x + x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

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

Add $x$ and $x$ .

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

Step 2

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

Tap for more steps...

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

$x^{2} + 2 x + 1$ $\neq$ ${(x - 1)}^{2}$

Since $x^{2} + 2 x + 1$ $\neq$ ${(x - 1)}^{2}$ , the function is not even.

The function is not even

Step 3

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

Tap for more steps...

Find $- f (x)$ .

Tap for more steps...

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$- f (x) = - ((x - 1) (x - 1))$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Tap for more steps...

Apply the distributive property.

$- f (x) = - (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$- f (x) = - (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$- f (x) = - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Simplify and combine like terms.

Tap for more steps...

Simplify each term.

Tap for more steps...

Multiply $x$ by $x$ .

$- f (x) = - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Move $- 1$ to the left of $x$ .

$- f (x) = - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$- f (x) = - (x^{2} - x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$- f (x) = - (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Apply the distributive property.

$- f (x) = - x^{2} - (- 2 x) - 1 \cdot 1$

Simplify.

Tap for more steps...

Multiply $- 2$ by $- 1$ .

$- f (x) = - x^{2} + 2 x - 1 \cdot 1$

Multiply $- 1$ by $1$ .

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

Since $x^{2} + 2 x + 1$ $\neq$ $- x^{2} + 2 x - 1$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5