Solve y=-2x^3 - ScanMath

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = - \frac{3 4 x}{2}

Evaluate

y = - 2 x^{3}

Interchange x and y

x = - 2 y^{3}

Swap the sides of the equation

- 2 y^{3} = x

Change the signs on both sides of the equation

2 y^{3} = - x

Divide both sides

\frac{2 y ^{3}}{2} = \frac{- x}{2}

Divide the numbers

y^{3} = \frac{- x}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

y^{3} = - \frac{x}{2}

Take the 3 -th root on both sides of the equation

3 y^{3} = 3 - \frac{x}{2}

Calculate

y = 3 - \frac{x}{2}

Simplify the root

More Steps

Evaluate

3 - \frac{x}{2}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{3 - x}{3 2}

Multiply by the Conjugate

\frac{3 - x \times 3 2 ^{2}}{3 2 \times 3 2 ^{2}}

Calculate

\frac{3 - x \times 3 2 ^{2}}{2}

Calculate

More Steps

Evaluate

3 - x \times 3 2^{2}

The product of roots with the same index is equal to the root of the product

3 - x \times 2^{2}

Calculate the product

3 - 2^{2} x

An odd root of a negative radicand is always a negative

- 3 2^{2} x

\frac{- 3 2 ^{2} x}{2}

Calculate

- \frac{3 2 ^{2} x}{2}

Calculate

- \frac{3 4 x}{2}

y = - \frac{3 4 x}{2}

Solution

f^{- 1} (x) = - \frac{3 4 x}{2}

Show Solution

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

Evaluate

y = - 2 x^{3}

To test if the graph of y = - 2 x^{3} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 2 (- x)^{3}

Simplify

More Steps

Evaluate

- 2 (- x)^{3}

Rewrite the expression

- 2 (- x^{3})

Multiply the numbers

2 x^{3}

- y = 2 x^{3}

Change the signs both sides

y = - 2 x^{3}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

x = - \frac{3 4 y}{2}

Evaluate

y = - 2 x^{3}

Swap the sides of the equation

- 2 x^{3} = y

Change the signs on both sides of the equation

2 x^{3} = - y

Divide both sides

\frac{2 x ^{3}}{2} = \frac{- y}{2}

Divide the numbers

x^{3} = \frac{- y}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x^{3} = - \frac{y}{2}

Take the 3 -th root on both sides of the equation

3 x^{3} = 3 - \frac{y}{2}

Calculate

x = 3 - \frac{y}{2}

Solution

More Steps

Evaluate

3 - \frac{y}{2}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{3 - y}{3 2}

Multiply by the Conjugate

\frac{3 - y \times 3 2 ^{2}}{3 2 \times 3 2 ^{2}}

Calculate

\frac{3 - y \times 3 2 ^{2}}{2}

Calculate

More Steps

Evaluate

3 - y \times 3 2^{2}

The product of roots with the same index is equal to the root of the product

3 - y \times 2^{2}

Calculate the product

3 - 2^{2} y

An odd root of a negative radicand is always a negative

- 3 2^{2} y

\frac{- 3 2 ^{2} y}{2}

Calculate

- \frac{3 2 ^{2} y}{2}

Calculate

- \frac{3 4 y}{2}

x = - \frac{3 4 y}{2}

Show Solution

Rewrite the equation

r = 0 r = - \frac{sin ( θ )}{2 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{2 cos ^{3} ( θ )}

Evaluate

y = - 2 x^{3}

Move the expression to the left side

y + 2 x^{3} = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

sin (θ) \times r + 2 (cos (θ) \times r)^{3} = 0

Factor the expression

2 cos^{3} (θ) \times r^{3} + sin (θ) \times r = 0

Factor the expression

r (2 cos^{3} (θ) \times r^{2} + sin (θ)) = 0

When the product of factors equals 0,at least one factor is 0

r = 0 2 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Solution

More Steps

Factor the expression

2 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Subtract the terms

2 cos^{3} (θ) \times r^{2} + sin (θ) - sin (θ) = 0 - sin (θ)

Evaluate

2 cos^{3} (θ) \times r^{2} = - sin (θ)

Divide the terms

r^{2} = - \frac{sin ( θ )}{2 cos ^{3} ( θ )}

Evaluate the power

r = \pm - \frac{sin ( θ )}{2 cos ^{3} ( θ )}

Separate into possible cases

r = - \frac{sin ( θ )}{2 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{2 cos ^{3} ( θ )}

r = 0 r = - \frac{sin ( θ )}{2 cos ^{3} ( θ )} r = - - \frac{sin ( θ )}{2 cos ^{3} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph