Solve y = -2x^5 - ScanMath

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = - \frac{5 16 x}{2}

Evaluate

y = - 2 x^{5}

Interchange x and y

x = - 2 y^{5}

Swap the sides of the equation

- 2 y^{5} = x

Change the signs on both sides of the equation

2 y^{5} = - x

Divide both sides

\frac{2 y ^{5}}{2} = \frac{- x}{2}

Divide the numbers

y^{5} = \frac{- x}{2}

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

y^{5} = - \frac{x}{2}

Take the 5 -th root on both sides of the equation

5 y^{5} = 5 - \frac{x}{2}

Calculate

y = 5 - \frac{x}{2}

Simplify the root

More Steps

Evaluate

5 - \frac{x}{2}

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

\frac{5 - x}{5 2}

Multiply by the Conjugate

\frac{5 - x \times 5 2 ^{4}}{5 2 \times 5 2 ^{4}}

Calculate

\frac{5 - x \times 5 2 ^{4}}{2}

Calculate

More Steps

Evaluate

5 - x \times 5 2^{4}

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

5 - x \times 2^{4}

Calculate the product

5 - 2^{4} x

An odd root of a negative radicand is always a negative

- 5 2^{4} x

\frac{- 5 2 ^{4} x}{2}

Calculate

- \frac{5 2 ^{4} x}{2}

Calculate

- \frac{5 16 x}{2}

y = - \frac{5 16 x}{2}

Solution

f^{- 1} (x) = - \frac{5 16 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^{5}

To test if the graph of y = - 2 x^{5} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 2 (- x)^{5}

Simplify

More Steps

Evaluate

- 2 (- x)^{5}

Rewrite the expression

- 2 (- x^{5})

Multiply the numbers

2 x^{5}

- y = 2 x^{5}

Change the signs both sides

y = - 2 x^{5}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

x = - \frac{5 16 y}{2}

Evaluate

y = - 2 x^{5}

Swap the sides of the equation

- 2 x^{5} = y

Change the signs on both sides of the equation

2 x^{5} = - y

Divide both sides

\frac{2 x ^{5}}{2} = \frac{- y}{2}

Divide the numbers

x^{5} = \frac{- y}{2}

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

x^{5} = - \frac{y}{2}

Take the 5 -th root on both sides of the equation

5 x^{5} = 5 - \frac{y}{2}

Calculate

x = 5 - \frac{y}{2}

Solution

More Steps

Evaluate

5 - \frac{y}{2}

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

\frac{5 - y}{5 2}

Multiply by the Conjugate

\frac{5 - y \times 5 2 ^{4}}{5 2 \times 5 2 ^{4}}

Calculate

\frac{5 - y \times 5 2 ^{4}}{2}

Calculate

More Steps

Evaluate

5 - y \times 5 2^{4}

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

5 - y \times 2^{4}

Calculate the product

5 - 2^{4} y

An odd root of a negative radicand is always a negative

- 5 2^{4} y

\frac{- 5 2 ^{4} y}{2}

Calculate

- \frac{5 2 ^{4} y}{2}

Calculate

- \frac{5 16 y}{2}

x = - \frac{5 16 y}{2}

Show Solution

Rewrite the equation

r = 0 r = 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )} r = - 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )}

Evaluate

y = - 2 x^{5}

Move the expression to the left side

y + 2 x^{5} = 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)^{5} = 0

Factor the expression

2 cos^{5} (θ) \times r^{5} + sin (θ) \times r = 0

Factor the expression

r (2 cos^{5} (θ) \times r^{4} + sin (θ)) = 0

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

r = 0 2 cos^{5} (θ) \times r^{4} + sin (θ) = 0

Solution

More Steps

Factor the expression

2 cos^{5} (θ) \times r^{4} + sin (θ) = 0

Subtract the terms

2 cos^{5} (θ) \times r^{4} + sin (θ) - sin (θ) = 0 - sin (θ)

Evaluate

2 cos^{5} (θ) \times r^{4} = - sin (θ)

Divide the terms

r^{4} = - \frac{sin ( θ )}{2 cos ^{5} ( θ )}

Evaluate the power

r = \pm 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )}

Separate into possible cases

r = 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )} r = - 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )}

r = 0 r = 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )} r = - 4 - \frac{sin ( θ )}{2 cos ^{5} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph