Algebra Examples

f (x) = 7 x^{3} - x

$f (x) = 7 x^{3} - x$

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Find

f (- x)

$f (- x)$ .

Tap for more steps...

Step 2.1

Find

f (- x)

$f (- x)$ by substituting

- x

$- x$ for all occurrence of

x

$x$ in

f (x)

$f (x)$ .

f (- x) = 7 {(- x)}^{3} - (- x)

$f (- x) = 7 {(- x)}^{3} - (- x)$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Apply the product rule to

- x

$- x$ .

f (- x) = 7 ({(- 1)}^{3} x^{3}) - (- x)

$f (- x) = 7 ({(- 1)}^{3} x^{3}) - (- x)$

Step 2.2.2

Raise

- 1

$- 1$ to the power of

3

$3$ .

f (- x) = 7 (- x^{3}) - (- x)

$f (- x) = 7 (- x^{3}) - (- x)$

Step 2.2.3

Multiply

- 1

$- 1$ by

7

$7$ .

f (- x) = - 7 x^{3} - (- x)

$f (- x) = - 7 x^{3} - (- x)$

Step 2.2.4

Multiply

- (- x)

$- (- x)$ .

Tap for more steps...

Step 2.2.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

f (- x) = - 7 x^{3} + 1 x

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

Step 2.2.4.2

Multiply

x

$x$ by

1

$1$ .

f (- x) = - 7 x^{3} + x

$f (- x) = - 7 x^{3} + x$

f (- x) = - 7 x^{3} + x

$f (- x) = - 7 x^{3} + x$

f (- x) = - 7 x^{3} + x

$f (- x) = - 7 x^{3} + x$

f (- x) = - 7 x^{3} + x

$f (- x) = - 7 x^{3} + x$

Step 3

A function is even if

f (- x) = f (x)

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

Tap for more steps...

Step 3.1

Check if

f (- x) = f (x)

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

Step 3.2

Since

- 7 x^{3} + x

$- 7 x^{3} + x$

\neq

$\neq$

7 x^{3} - x

$7 x^{3} - x$ , the function is not even.

The function is not even

Step 4

A function is odd if

f (- x) = - f (x)

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

Tap for more steps...

Step 4.1

Find

- f (x)

$- f (x)$ .

Tap for more steps...

Step 4.1.1

Multiply

7 x^{3} - x

$7 x^{3} - x$ by

- 1

$- 1$ .

- f (x) = - (7 x^{3} - x)

$- f (x) = - (7 x^{3} - x)$

Step 4.1.2

Apply the distributive property.

- f (x) = - (7 x^{3}) + x

$- f (x) = - (7 x^{3}) + x$

Step 4.1.3

Multiply

7

$7$ by

- 1

$- 1$ .

- f (x) = - 7 x^{3} + x

$- f (x) = - 7 x^{3} + x$

- f (x) = - 7 x^{3} + x

$- f (x) = - 7 x^{3} + x$

Step 4.2

Since

- 7 x^{3} + x = - 7 x^{3} + x

$- 7 x^{3} + x = - 7 x^{3} + x$ , the function is odd.

The function is odd

Step 5

Since the function is odd, it is symmetric about the origin.

Origin Symmetry

Step 6

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Step 7

Determine the symmetry of the function.

Origin symmetry

Step 8