Solve y=x^211x 18

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = \frac{3 39204 x}{198}

Evaluate

y = x^{2} \times 11 x \times 18

Simplify

More Steps

Evaluate

x^{2} \times 11 x \times 18

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 11 \times 18

Add the numbers

x^{3} \times 11 \times 18

Multiply the terms

x^{3} \times 198

Use the commutative property to reorder the terms

198 x^{3}

y = 198 x^{3}

Interchange x and y

x = 198 y^{3}

Swap the sides of the equation

198 y^{3} = x

Divide both sides

\frac{198 y ^{3}}{198} = \frac{x}{198}

Divide the numbers

y^{3} = \frac{x}{198}

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

3 y^{3} = 3 \frac{x}{198}

Calculate

y = 3 \frac{x}{198}

Simplify the root

More Steps

Evaluate

3 \frac{x}{198}

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

\frac{3 x}{3 198}

Multiply by the Conjugate

\frac{3 x \times 3 19 8 ^{2}}{3 198 \times 3 19 8 ^{2}}

Calculate

\frac{3 x \times 3 19 8 ^{2}}{198}

Calculate

More Steps

Evaluate

3 x \times 3 19 8^{2}

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

3 x \times 19 8^{2}

Calculate the product

3 19 8^{2} x

\frac{3 19 8 ^{2} x}{198}

Calculate

\frac{3 39204 x}{198}

y = \frac{3 39204 x}{198}

Solution

f^{- 1} (x) = \frac{3 39204 x}{198}

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 = x^{2} 11 x 18

Simplify the expression

y = 198 x^{3}

To test if the graph of y = 198 x^{3} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 198 (- x)^{3}

Simplify

More Steps

Evaluate

198 (- x)^{3}

Rewrite the expression

198 (- x^{3})

Multiply the numbers

- 198 x^{3}

- y = - 198 x^{3}

Change the signs both sides

y = 198 x^{3}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{3 39204 y}{198}

Evaluate

y = x^{2} \times 11 x \times 18

Simplify

More Steps

Evaluate

x^{2} \times 11 x \times 18

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 11 \times 18

Add the numbers

x^{3} \times 11 \times 18

Multiply the terms

x^{3} \times 198

Use the commutative property to reorder the terms

198 x^{3}

y = 198 x^{3}

Swap the sides of the equation

198 x^{3} = y

Divide both sides

\frac{198 x ^{3}}{198} = \frac{y}{198}

Divide the numbers

x^{3} = \frac{y}{198}

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

3 x^{3} = 3 \frac{y}{198}

Calculate

x = 3 \frac{y}{198}

Solution

More Steps

Evaluate

3 \frac{y}{198}

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

\frac{3 y}{3 198}

Multiply by the Conjugate

\frac{3 y \times 3 19 8 ^{2}}{3 198 \times 3 19 8 ^{2}}

Calculate

\frac{3 y \times 3 19 8 ^{2}}{198}

Calculate

More Steps

Evaluate

3 y \times 3 19 8^{2}

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

3 y \times 19 8^{2}

Calculate the product

3 19 8^{2} y

\frac{3 19 8 ^{2} y}{198}

Calculate

\frac{3 39204 y}{198}

x = \frac{3 39204 y}{198}

Show Solution

Rewrite the equation

r = 0 r = \frac{sin ( θ )}{198 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{198 cos ^{3} ( θ )}

Evaluate

y = x^{2} \times 11 x \times 18

Simplify

More Steps

Evaluate

x^{2} \times 11 x \times 18

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 11 \times 18

Add the numbers

x^{3} \times 11 \times 18

Multiply the terms

x^{3} \times 198

Use the commutative property to reorder the terms

198 x^{3}

y = 198 x^{3}

Move the expression to the left side

y - 198 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 - 198 (cos (θ) \times r)^{3} = 0

Factor the expression

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

Factor the expression

r (- 198 cos^{3} (θ) \times r^{2} + sin (θ)) = 0

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

r = 0 - 198 cos^{3} (θ) \times r^{2} + sin (θ) = 0

Solution

More Steps

Factor the expression

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

Subtract the terms

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

Evaluate

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

Divide the terms

r^{2} = \frac{sin ( θ )}{198 cos ^{3} ( θ )}

Evaluate the power

r = \pm \frac{sin ( θ )}{198 cos ^{3} ( θ )}

Separate into possible cases

r = \frac{sin ( θ )}{198 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{198 cos ^{3} ( θ )}

r = 0 r = \frac{sin ( θ )}{198 cos ^{3} ( θ )} r = - \frac{sin ( θ )}{198 cos ^{3} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph