Solve y = 15x^3500

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = \frac{3 3600 x}{300}

Evaluate

y = 15 x^{3} \times 500

Simplify

y = 7500 x^{3}

Interchange x and y

x = 7500 y^{3}

Swap the sides of the equation

7500 y^{3} = x

Divide both sides

\frac{7500 y ^{3}}{7500} = \frac{x}{7500}

Divide the numbers

y^{3} = \frac{x}{7500}

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

3 y^{3} = 3 \frac{x}{7500}

Calculate

y = 3 \frac{x}{7500}

Simplify the root

More Steps

Evaluate

3 \frac{x}{7500}

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

\frac{3 x}{3 7500}

Simplify the radical expression

More Steps

Evaluate

37500

Write the expression as a product where the root of one of the factors can be evaluated

3 125 \times 60

Write the number in exponential form with the base of 5

3 5^{3} \times 60

The root of a product is equal to the product of the roots of each factor

3 5^{3} \times 360

Reduce the index of the radical and exponent with 3

5360

\frac{3 x}{5 3 60}

Multiply by the Conjugate

\frac{3 x \times 3 6 0 ^{2}}{5 3 60 \times 3 6 0 ^{2}}

Calculate

\frac{3 x \times 3 6 0 ^{2}}{5 \times 60}

Calculate

More Steps

Evaluate

3 x \times 3 6 0^{2}

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

3 x \times 6 0^{2}

Calculate the product

3 6 0^{2} x

\frac{3 6 0 ^{2} x}{5 \times 60}

Calculate

\frac{3 6 0 ^{2} x}{300}

Calculate

\frac{3 3600 x}{300}

y = \frac{3 3600 x}{300}

Solution

f^{- 1} (x) = \frac{3 3600 x}{300}

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 = 15 x^{3} 500

Simplify the expression

y = 7500 x^{3}

To test if the graph of y = 7500 x^{3} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 7500 (- x)^{3}

Simplify

More Steps

Evaluate

7500 (- x)^{3}

Rewrite the expression

7500 (- x^{3})

Multiply the numbers

- 7500 x^{3}

- y = - 7500 x^{3}

Change the signs both sides

y = 7500 x^{3}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{3 3600 y}{300}

Evaluate

y = 15 x^{3} \times 500

Simplify

y = 7500 x^{3}

Swap the sides of the equation

7500 x^{3} = y

Divide both sides

\frac{7500 x ^{3}}{7500} = \frac{y}{7500}

Divide the numbers

x^{3} = \frac{y}{7500}

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

3 x^{3} = 3 \frac{y}{7500}

Calculate

x = 3 \frac{y}{7500}

Solution

More Steps

Evaluate

3 \frac{y}{7500}

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

\frac{3 y}{3 7500}

Simplify the radical expression

More Steps

Evaluate

37500

Write the expression as a product where the root of one of the factors can be evaluated

3 125 \times 60

Write the number in exponential form with the base of 5

3 5^{3} \times 60

The root of a product is equal to the product of the roots of each factor

3 5^{3} \times 360

Reduce the index of the radical and exponent with 3

5360

\frac{3 y}{5 3 60}

Multiply by the Conjugate

\frac{3 y \times 3 6 0 ^{2}}{5 3 60 \times 3 6 0 ^{2}}

Calculate

\frac{3 y \times 3 6 0 ^{2}}{5 \times 60}

Calculate

More Steps

Evaluate

3 y \times 3 6 0^{2}

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

3 y \times 6 0^{2}

Calculate the product

3 6 0^{2} y

\frac{3 6 0 ^{2} y}{5 \times 60}

Calculate

\frac{3 6 0 ^{2} y}{300}

Calculate

\frac{3 3600 y}{300}

x = \frac{3 3600 y}{300}

Show Solution

Rewrite the equation

r = 0 r = \frac{sin ( θ )}{7500 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{7500 cos ^{3} ( θ )}

Evaluate

y = 15 x^{3} \times 500

Simplify

y = 7500 x^{3}

Move the expression to the left side

y - 7500 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 - 7500 (cos (θ) \times r)^{3} = 0

Factor the expression

- 7500 cos^{3} (θ) \times r^{3} + sin (θ) \times r = 0

Factor the expression

r (- 7500 cos^{3} (θ) \times r^{2} + sin (θ)) = 0

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

r = 0 - 7500 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Solution

More Steps

Factor the expression

- 7500 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Subtract the terms

- 7500 cos^{3} (θ) \times r^{2} + sin (θ) - sin (θ) = 0 - sin (θ)

Evaluate

- 7500 cos^{3} (θ) \times r^{2} = - sin (θ)

Divide the terms

r^{2} = \frac{sin ( θ )}{7500 cos ^{3} ( θ )}

Evaluate the power

r = \pm \frac{sin ( θ )}{7500 cos ^{3} ( θ )}

Separate into possible cases

r = \frac{sin ( θ )}{7500 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{7500 cos ^{3} ( θ )}

r = 0 r = \frac{sin ( θ )}{7500 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{7500 cos ^{3} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph