Determine if Odd, Even, or Neither x square root of 7-x^2

$x \sqrt{7 - x^{2}}$

Step 1

Write $x \sqrt{7 - x^{2}}$ as a function.

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

Step 2

Find $f (- x)$ .

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

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

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

$f (- x) = - x \sqrt{7 + {(- 1)}^{2} \cdot (- 1 x^{2})}$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

$f (- x) = - x \sqrt{7 + {(- 1)}^{2} \cdot ((- 1) x^{2})}$

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

$f (- x) = - x \sqrt{7 + {(- 1)}^{2 + 1} x^{2}}$

Add $2$ and $1$ .

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

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

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

Step 3

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

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

Since $- x \sqrt{7 - x^{2}}$ $\neq$ $x \sqrt{7 - x^{2}}$ , the function is not even.

The function is not even

Step 4

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

Remove parentheses.

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

Since $- x \sqrt{7 - x^{2}} = - x \sqrt{7 - x^{2}}$ , the function is odd.

The function is odd

Step 5