Solve y = 5x^34x^2 - 3x^5

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = \frac{5 1 7 ^{4} x}{17}

Evaluate

y = 5 x^{3} \times 4 x^{2} - 3 x^{5}

Simplify

More Steps

Evaluate

5 x^{3} \times 4 x^{2} - 3 x^{5}

Multiply

More Steps

Multiply the terms

5 x^{3} \times 4 x^{2}

Multiply the terms

20 x^{3} \times x^{2}

Multiply the terms with the same base by adding their exponents

20 x^{3 + 2}

Add the numbers

20 x^{5}

20 x^{5} - 3 x^{5}

Collect like terms by calculating the sum or difference of their coefficients

(20 - 3) x^{5}

Subtract the numbers

17 x^{5}

y = 17 x^{5}

Interchange x and y

x = 17 y^{5}

Swap the sides of the equation

17 y^{5} = x

Divide both sides

\frac{17 y ^{5}}{17} = \frac{x}{17}

Divide the numbers

y^{5} = \frac{x}{17}

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

5 y^{5} = 5 \frac{x}{17}

Calculate

y = 5 \frac{x}{17}

Simplify the root

More Steps

Evaluate

5 \frac{x}{17}

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

\frac{5 x}{5 17}

Multiply by the Conjugate

\frac{5 x \times 5 1 7 ^{4}}{5 17 \times 5 1 7 ^{4}}

Calculate

\frac{5 x \times 5 1 7 ^{4}}{17}

Calculate

More Steps

Evaluate

5 x \times 5 1 7^{4}

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

5 x \times 1 7^{4}

Calculate the product

5 1 7^{4} x

\frac{5 1 7 ^{4} x}{17}

y = \frac{5 1 7 ^{4} x}{17}

Solution

f^{- 1} (x) = \frac{5 1 7 ^{4} x}{17}

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 = 5 x^{3} 4 x^{2} - 3 x^{5}

Simplify the expression

y = 17 x^{5}

To test if the graph of y = 17 x^{5} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 17 (- x)^{5}

Simplify

More Steps

Evaluate

17 (- x)^{5}

Rewrite the expression

17 (- x^{5})

Multiply the numbers

- 17 x^{5}

- y = - 17 x^{5}

Change the signs both sides

y = 17 x^{5}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{5 1 7 ^{4} y}{17}

Evaluate

y = 5 x^{3} \times 4 x^{2} - 3 x^{5}

Simplify

More Steps

Evaluate

5 x^{3} \times 4 x^{2} - 3 x^{5}

Multiply

More Steps

Multiply the terms

5 x^{3} \times 4 x^{2}

Multiply the terms

20 x^{3} \times x^{2}

Multiply the terms with the same base by adding their exponents

20 x^{3 + 2}

Add the numbers

20 x^{5}

20 x^{5} - 3 x^{5}

Collect like terms by calculating the sum or difference of their coefficients

(20 - 3) x^{5}

Subtract the numbers

17 x^{5}

y = 17 x^{5}

Swap the sides of the equation

17 x^{5} = y

Divide both sides

\frac{17 x ^{5}}{17} = \frac{y}{17}

Divide the numbers

x^{5} = \frac{y}{17}

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

5 x^{5} = 5 \frac{y}{17}

Calculate

x = 5 \frac{y}{17}

Solution

More Steps

Evaluate

5 \frac{y}{17}

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

\frac{5 y}{5 17}

Multiply by the Conjugate

\frac{5 y \times 5 1 7 ^{4}}{5 17 \times 5 1 7 ^{4}}

Calculate

\frac{5 y \times 5 1 7 ^{4}}{17}

Calculate

More Steps

Evaluate

5 y \times 5 1 7^{4}

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

5 y \times 1 7^{4}

Calculate the product

5 1 7^{4} y

\frac{5 1 7 ^{4} y}{17}

x = \frac{5 1 7 ^{4} y}{17}

Show Solution

Rewrite the equation

r = 0 r = 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )} r = - 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )}

Evaluate

y = 5 x^{3} \times 4 x^{2} - 3 x^{5}

Simplify

More Steps

Evaluate

5 x^{3} \times 4 x^{2} - 3 x^{5}

Multiply

More Steps

Multiply the terms

5 x^{3} \times 4 x^{2}

Multiply the terms

20 x^{3} \times x^{2}

Multiply the terms with the same base by adding their exponents

20 x^{3 + 2}

Add the numbers

20 x^{5}

20 x^{5} - 3 x^{5}

Collect like terms by calculating the sum or difference of their coefficients

(20 - 3) x^{5}

Subtract the numbers

17 x^{5}

y = 17 x^{5}

Move the expression to the left side

y - 17 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 - 17 (cos (θ) \times r)^{5} = 0

Factor the expression

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

Factor the expression

r (- 17 cos^{5} (θ) \times r^{4} + sin (θ)) = 0

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

r = 0 - 17 cos^{5} (θ) \times r^{4} + sin (θ) = 0

Solution

More Steps

Factor the expression

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

Subtract the terms

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

Evaluate

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

Divide the terms

r^{4} = \frac{sin ( θ )}{17 cos ^{5} ( θ )}

Evaluate the power

r = \pm 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )}

Separate into possible cases

r = 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )} r = - 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )}

r = 0 r = 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )} r = - 4 \frac{sin ( θ )}{17 cos ^{5} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph