Finite Math Examples

$g (x) = x^{2} (4 - x)$ , $h (x) = 7 x - 1$

Step 1

Set up the composite result function.

$g (h (x))$

Step 2

Evaluate $g (7 x - 1)$ by substituting in the value of $h$ into $g$ .

$g (7 x - 1) = {(7 x - 1)}^{2} (4 - (7 x - 1))$

Step 3

Rewrite ${(7 x - 1)}^{2}$ as $(7 x - 1) (7 x - 1)$ .

$g (7 x - 1) = (7 x - 1) (7 x - 1) (4 - (7 x - 1))$

Step 4

Expand $(7 x - 1) (7 x - 1)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$g (7 x - 1) = (7 x (7 x - 1) - 1 (7 x - 1)) (4 - (7 x - 1))$

Step 4.2

Apply the distributive property.

$g (7 x - 1) = (7 x (7 x) + 7 x \cdot - 1 - 1 (7 x - 1)) (4 - (7 x - 1))$

Step 4.3

Apply the distributive property.

$g (7 x - 1) = (7 x (7 x) + 7 x \cdot - 1 - 1 (7 x) - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$g (7 x - 1) = (7 \cdot (7 x \cdot x) + 7 x \cdot - 1 - 1 (7 x) - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.2.1

Move $x$ .

$g (7 x - 1) = (7 \cdot (7 (x \cdot x)) + 7 x \cdot - 1 - 1 (7 x) - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5.1.2.2

Multiply $x$ by $x$ .

$g (7 x - 1) = (7 \cdot (7 x^{2}) + 7 x \cdot - 1 - 1 (7 x) - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5.1.3

Multiply $7$ by $7$ .

$g (7 x - 1) = (49 x^{2} + 7 x \cdot - 1 - 1 (7 x) - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5.1.4

Multiply $- 1$ by $7$ .

$g (7 x - 1) = (49 x^{2} - 7 x - 1 (7 x) - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5.1.5

Multiply $7$ by $- 1$ .

$g (7 x - 1) = (49 x^{2} - 7 x - 7 x - 1 \cdot - 1) (4 - (7 x - 1))$

Step 5.1.6

Multiply $- 1$ by $- 1$ .

$g (7 x - 1) = (49 x^{2} - 7 x - 7 x + 1) (4 - (7 x - 1))$

Step 5.2

Subtract $7 x$ from $- 7 x$ .

$g (7 x - 1) = (49 x^{2} - 14 x + 1) (4 - (7 x - 1))$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

$g (7 x - 1) = (49 x^{2} - 14 x + 1) (4 - (7 x) + 1)$

Step 6.1.2

Multiply $7$ by $- 1$ .

$g (7 x - 1) = (49 x^{2} - 14 x + 1) (4 - 7 x + 1)$

Step 6.1.3

Multiply $- 1$ by $- 1$ .

$g (7 x - 1) = (49 x^{2} - 14 x + 1) (4 - 7 x + 1)$

Step 6.2

Add $4$ and $1$ .

$g (7 x - 1) = (49 x^{2} - 14 x + 1) (- 7 x + 5)$

Step 7

Expand $(49 x^{2} - 14 x + 1) (- 7 x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$g (7 x - 1) = 49 x^{2} (- 7 x) + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8

Simplify terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite using the commutative property of multiplication.

$g (7 x - 1) = 49 \cdot (- 7 x^{2} x) + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.1.2.1

Move $x$ .

$g (7 x - 1) = 49 \cdot (- 7 (x \cdot x^{2})) + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.2.2

Multiply $x$ by $x^{2}$ .

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Step 8.1.2.2.1

Raise $x$ to the power of $1$ .

$g (7 x - 1) = 49 \cdot (- 7 (x \cdot x^{2})) + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$g (7 x - 1) = 49 \cdot (- 7 x^{1 + 2}) + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.2.3

Add $1$ and $2$ .

$g (7 x - 1) = 49 \cdot (- 7 x^{3}) + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.3

Multiply $49$ by $- 7$ .

$g (7 x - 1) = - 343 x^{3} + 49 x^{2} \cdot 5 - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.4

Multiply $5$ by $49$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} - 14 x (- 7 x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.5

Rewrite using the commutative property of multiplication.

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} - 14 \cdot (- 7 x \cdot x) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.6

Multiply $x$ by $x$ by adding the exponents.

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Step 8.1.6.1

Move $x$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} - 14 \cdot (- 7 (x \cdot x)) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.6.2

Multiply $x$ by $x$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} - 14 \cdot (- 7 x^{2}) - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.7

Multiply $- 14$ by $- 7$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} + 98 x^{2} - 14 x \cdot 5 + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.8

Multiply $5$ by $- 14$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} + 98 x^{2} - 70 x + 1 (- 7 x) + 1 \cdot 5$

Step 8.1.9

Multiply $- 7 x$ by $1$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} + 98 x^{2} - 70 x - 7 x + 1 \cdot 5$

Step 8.1.10

Multiply $5$ by $1$ .

$g (7 x - 1) = - 343 x^{3} + 245 x^{2} + 98 x^{2} - 70 x - 7 x + 5$

Step 8.2

Simplify by adding terms.

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Step 8.2.1

Add $245 x^{2}$ and $98 x^{2}$ .

$g (7 x - 1) = - 343 x^{3} + 343 x^{2} - 70 x - 7 x + 5$

Step 8.2.2

Subtract $7 x$ from $- 70 x$ .

$g (7 x - 1) = - 343 x^{3} + 343 x^{2} - 77 x + 5$