Solve y= -2x^2 12x - 19

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = - \frac{3 9 x + 171}{6}

Evaluate

y = - 2 x^{2} \times 12 x - 19

Simplify

More Steps

Evaluate

- 2 x^{2} \times 12 x - 19

Multiply

More Steps

Evaluate

- 2 x^{2} \times 12 x

Multiply the terms

- 24 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 24 x^{2 + 1}

Add the numbers

- 24 x^{3}

- 24 x^{3} - 19

y = - 24 x^{3} - 19

Interchange x and y

x = - 24 y^{3} - 19

Swap the sides of the equation

- 24 y^{3} - 19 = x

Move the constant to the right-hand side and change its sign

- 24 y^{3} = x + 19

Change the signs on both sides of the equation

24 y^{3} = - x - 19

Divide both sides

\frac{24 y ^{3}}{24} = \frac{- x - 19}{24}

Divide the numbers

y^{3} = \frac{- x - 19}{24}

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

y^{3} = - \frac{x + 19}{24}

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

3 y^{3} = 3 - \frac{x + 19}{24}

Calculate

y = 3 - \frac{x + 19}{24}

Simplify the root

More Steps

Evaluate

3 - \frac{x + 19}{24}

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

\frac{3 - x - 19}{3 24}

Simplify the radical expression

\frac{- 3 x + 19}{3 24}

Simplify the radical expression

More Steps

Evaluate

324

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

3 8 \times 3

Write the number in exponential form with the base of 2

3 2^{3} \times 3

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

3 2^{3} \times 33

Reduce the index of the radical and exponent with 3

233

- \frac{3 x + 19}{2 3 3}

Multiply by the Conjugate

- \frac{3 x + 19 \times 3 3 ^{2}}{2 3 3 \times 3 3 ^{2}}

Calculate

- \frac{3 x + 19 \times 3 3 ^{2}}{2 \times 3}

Calculate

More Steps

Evaluate

3 x + 19 \times 3 3^{2}

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

3 (x + 19) \times 3^{2}

Calculate the product

3 9 x + 171

- \frac{3 9 x + 171}{2 \times 3}

Calculate

- \frac{3 9 x + 171}{6}

y = - \frac{3 9 x + 171}{6}

Solution

f^{- 1} (x) = - \frac{3 9 x + 171}{6}

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

Not symmetry with respect to the origin

Evaluate

y = - 2 x^{2} 12 x - 19

Simplify the expression

y = - 24 x^{3} - 19

To test if the graph of y = - 24 x^{3} - 19 is symmetry with respect to the origin,substitute -x for x and -y for y

- y = - 24 (- x)^{3} - 19

Simplify

More Steps

Evaluate

- 24 (- x)^{3} - 19

Multiply the terms

More Steps

Evaluate

- 24 (- x)^{3}

Rewrite the expression

- 24 (- x^{3})

Multiply the numbers

24 x^{3}

24 x^{3} - 19

- y = 24 x^{3} - 19

Change the signs both sides

y = - 24 x^{3} + 19

Solution

Not symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = - \frac{3 9 y + 171}{6}

Evaluate

y = - 2 x^{2} \times 12 x - 19

Simplify

More Steps

Evaluate

- 2 x^{2} \times 12 x - 19

Multiply

More Steps

Evaluate

- 2 x^{2} \times 12 x

Multiply the terms

- 24 x^{2} \times x

Multiply the terms with the same base by adding their exponents

- 24 x^{2 + 1}

Add the numbers

- 24 x^{3}

- 24 x^{3} - 19

y = - 24 x^{3} - 19

Swap the sides of the equation

- 24 x^{3} - 19 = y

Move the constant to the right-hand side and change its sign

- 24 x^{3} = y + 19

Change the signs on both sides of the equation

24 x^{3} = - y - 19

Divide both sides

\frac{24 x ^{3}}{24} = \frac{- y - 19}{24}

Divide the numbers

x^{3} = \frac{- y - 19}{24}

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

x^{3} = - \frac{y + 19}{24}

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

3 x^{3} = 3 - \frac{y + 19}{24}

Calculate

x = 3 - \frac{y + 19}{24}

Solution

More Steps

Evaluate

3 - \frac{y + 19}{24}

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

\frac{3 - y - 19}{3 24}

Simplify the radical expression

\frac{- 3 y + 19}{3 24}

Simplify the radical expression

More Steps

Evaluate

324

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

3 8 \times 3

Write the number in exponential form with the base of 2

3 2^{3} \times 3

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

3 2^{3} \times 33

Reduce the index of the radical and exponent with 3

233

- \frac{3 y + 19}{2 3 3}

Multiply by the Conjugate

- \frac{3 y + 19 \times 3 3 ^{2}}{2 3 3 \times 3 3 ^{2}}

Calculate

- \frac{3 y + 19 \times 3 3 ^{2}}{2 \times 3}

Calculate

More Steps

Evaluate

3 y + 19 \times 3 3^{2}

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

3 (y + 19) \times 3^{2}

Calculate the product

3 9 y + 171

- \frac{3 9 y + 171}{2 \times 3}

Calculate

- \frac{3 9 y + 171}{6}

x = - \frac{3 9 y + 171}{6}

Show Solution

Function

Testing for symmetry

Solve the equation

Graph