Solve h(t)=-3t^2 12t 96

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

h^{- 1} (t) = - \frac{3 4 t}{24}

Evaluate

h (t) = - 3 t^{2} \times 12 t \times 96

Simplify

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Evaluate

- 3 t^{2} \times 12 t \times 96

Multiply the terms

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Evaluate

3 \times 12 \times 96

Multiply the terms

36 \times 96

Multiply the numbers

3456

- 3456 t^{2} \times t

Multiply the terms with the same base by adding their exponents

- 3456 t^{2 + 1}

Add the numbers

- 3456 t^{3}

h (t) = - 3456 t^{3}

In the equation for h (t),replace h (t) with y

y = - 3456 t^{3}

Interchange t and y

t = - 3456 y^{3}

Swap the sides of the equation

- 3456 y^{3} = t

Change the signs on both sides of the equation

3456 y^{3} = - t

Divide both sides

\frac{3456 y ^{3}}{3456} = \frac{- t}{3456}

Divide the numbers

y^{3} = \frac{- t}{3456}

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

y^{3} = - \frac{t}{3456}

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

3 y^{3} = 3 - \frac{t}{3456}

Calculate

y = 3 - \frac{t}{3456}

Simplify the root

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Evaluate

3 - \frac{t}{3456}

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

\frac{3 - t}{3 3456}

Simplify the radical expression

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Evaluate

33456

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

3 1728 \times 2

Write the number in exponential form with the base of 12

3 1 2^{3} \times 2

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

3 1 2^{3} \times 32

Reduce the index of the radical and exponent with 3

1232

\frac{3 - t}{12 3 2}

Multiply by the Conjugate

\frac{3 - t \times 3 2 ^{2}}{12 3 2 \times 3 2 ^{2}}

Calculate

\frac{3 - t \times 3 2 ^{2}}{12 \times 2}

Calculate

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Evaluate

3 - t \times 3 2^{2}

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

3 - t \times 2^{2}

Calculate the product

3 - 2^{2} t

An odd root of a negative radicand is always a negative

- 3 2^{2} t

\frac{- 3 2 ^{2} t}{12 \times 2}

Calculate

\frac{- 3 2 ^{2} t}{24}

Calculate

- \frac{3 2 ^{2} t}{24}

Calculate

- \frac{3 4 t}{24}

y = - \frac{3 4 t}{24}

Solution

h^{- 1} (t) = - \frac{3 4 t}{24}

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