Determine if Odd, Even, or Neither g(x)=(x-1)^2+1

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

Step 1

Simplify.

Simplify each term.

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

$g (x) = (x - 1) (x - 1) + 1$

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

Apply the distributive property.

$g (x) = x (x - 1) - 1 (x - 1) + 1$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Add $1$ and $1$ .

$g (x) = x^{2} - 2 x + 2$

Step 2

Find $f (- x)$ .

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

$g (- x) = {(- x)}^{2} - 2 (- x) + 2$

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

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

$g (- x) = x^{2} - 2 (- x) + 2$

Multiply $- 1$ by $- 2$ .

$g (- x) = x^{2} + 2 x + 2$

Step 3

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

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

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

The function is not even

Step 4

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

Find $- f (x)$ .

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

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

Apply the distributive property.

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

Simplify.

Multiply $- 2$ by $- 1$ .

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

Multiply $- 1$ by $2$ .

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

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

The function is not odd

Step 5

The function is neither odd nor even

Step 6

Determine if Odd, Even, or Neither g(x)=(x-1)^2+1

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