Precalculus Examples

$h (a) = - 2 a^{3} + 3 a$ $g (a) = a - 3$ find $h (g (- 1))$

Step 1

Set up the composite result function.

$h (g (a))$

Step 2

Evaluate $h (a - 3)$ by substituting in the value of $g$ into $h$ .

$h (a - 3) = - 2 {(a - 3)}^{3} + 3 (a - 3)$

Step 3

Simplify each term.

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Step 3.1

Use the Binomial Theorem.

$h (a - 3) = - 2 (a^{3} + 3 a^{2} \cdot - 3 + 3 a {(- 3)}^{2} + {(- 3)}^{3}) + 3 (a - 3)$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $- 3$ by $3$ .

$h (a - 3) = - 2 (a^{3} - 9 a^{2} + 3 a {(- 3)}^{2} + {(- 3)}^{3}) + 3 (a - 3)$

Step 3.2.2

Raise $- 3$ to the power of $2$ .

$h (a - 3) = - 2 (a^{3} - 9 a^{2} + 3 a \cdot 9 + {(- 3)}^{3}) + 3 (a - 3)$

Step 3.2.3

Multiply $9$ by $3$ .

$h (a - 3) = - 2 (a^{3} - 9 a^{2} + 27 a + {(- 3)}^{3}) + 3 (a - 3)$

Step 3.2.4

Raise $- 3$ to the power of $3$ .

$h (a - 3) = - 2 (a^{3} - 9 a^{2} + 27 a - 27) + 3 (a - 3)$

Step 3.3

Apply the distributive property.

$h (a - 3) = - 2 a^{3} - 2 (- 9 a^{2}) - 2 (27 a) - 2 \cdot - 27 + 3 (a - 3)$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $- 9$ by $- 2$ .

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 2 (27 a) - 2 \cdot - 27 + 3 (a - 3)$

Step 3.4.2

Multiply $27$ by $- 2$ .

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 54 a - 2 \cdot - 27 + 3 (a - 3)$

Step 3.4.3

Multiply $- 2$ by $- 27$ .

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 54 a + 54 + 3 (a - 3)$

Step 3.5

Apply the distributive property.

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 54 a + 54 + 3 a + 3 \cdot - 3$

Step 3.6

Multiply $3$ by $- 3$ .

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 54 a + 54 + 3 a - 9$

Step 4

Simplify by adding terms.

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Step 4.1

Add $- 54 a$ and $3 a$ .

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 51 a + 54 - 9$

Step 4.2

Subtract $9$ from $54$ .

$h (a - 3) = - 2 a^{3} + 18 a^{2} - 51 a + 45$

Step 5

Evaluate the result function by replacing $a$ with $- 1$ in the function.

$- 2 {(- 1)}^{3} + 18 {(- 1)}^{2} - 51 \cdot - 1 + 45$

Step 6

Simplify the evaluated function.

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Step 6.1

Simplify each term.

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Step 6.1.1

Raise $- 1$ to the power of $3$ .

$- 2 \cdot - 1 + 18 {(- 1)}^{2} - 51 \cdot - 1 + 45$

Step 6.1.2

Multiply $- 2$ by $- 1$ .

$2 + 18 {(- 1)}^{2} - 51 \cdot - 1 + 45$

Step 6.1.3

Raise $- 1$ to the power of $2$ .

$2 + 18 \cdot 1 - 51 \cdot - 1 + 45$

Step 6.1.4

Multiply $18$ by $1$ .

$2 + 18 - 51 \cdot - 1 + 45$

Step 6.1.5

Multiply $- 51$ by $- 1$ .

$2 + 18 + 51 + 45$

Step 6.2

Simplify by adding numbers.

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Step 6.2.1

Add $2$ and $18$ .

$20 + 51 + 45$

Step 6.2.2

Add $20$ and $51$ .

$71 + 45$

Step 6.2.3

Add $71$ and $45$ .

$116$