Calculus Examples

$f (x) = 4 x - 2$ , $g (x) = 3 x^{2}$ , $g (f (\frac{3}{2}))$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (4 x - 2)$ by substituting in the value of $f$ into $g$ .

$g (4 x - 2) = 3 {(4 x - 2)}^{2}$

Step 3

Rewrite ${(4 x - 2)}^{2}$ as $(4 x - 2) (4 x - 2)$ .

$g (4 x - 2) = 3 ((4 x - 2) (4 x - 2))$

Step 4

Expand $(4 x - 2) (4 x - 2)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$g (4 x - 2) = 3 (4 x (4 x - 2) - 2 (4 x - 2))$

Step 4.2

Apply the distributive property.

$g (4 x - 2) = 3 (4 x (4 x) + 4 x \cdot - 2 - 2 (4 x - 2))$

Step 4.3

Apply the distributive property.

$g (4 x - 2) = 3 (4 x (4 x) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2)$

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.

$g (4 x - 2) = 3 (4 \cdot (4 x \cdot x) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2)$

Step 5.1.2

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

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Step 5.1.2.1

Move $x$ .

$g (4 x - 2) = 3 (4 \cdot (4 (x \cdot x)) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2)$

Step 5.1.2.2

Multiply $x$ by $x$ .

$g (4 x - 2) = 3 (4 \cdot (4 x^{2}) + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2)$

Step 5.1.3

Multiply $4$ by $4$ .

$g (4 x - 2) = 3 (16 x^{2} + 4 x \cdot - 2 - 2 (4 x) - 2 \cdot - 2)$

Step 5.1.4

Multiply $- 2$ by $4$ .

$g (4 x - 2) = 3 (16 x^{2} - 8 x - 2 (4 x) - 2 \cdot - 2)$

Step 5.1.5

Multiply $4$ by $- 2$ .

$g (4 x - 2) = 3 (16 x^{2} - 8 x - 8 x - 2 \cdot - 2)$

Step 5.1.6

Multiply $- 2$ by $- 2$ .

$g (4 x - 2) = 3 (16 x^{2} - 8 x - 8 x + 4)$

Step 5.2

Subtract $8 x$ from $- 8 x$ .

$g (4 x - 2) = 3 (16 x^{2} - 16 x + 4)$

Step 6

Apply the distributive property.

$g (4 x - 2) = 3 (16 x^{2}) + 3 (- 16 x) + 3 \cdot 4$

Step 7

Simplify.

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Step 7.1

Multiply $16$ by $3$ .

$g (4 x - 2) = 48 x^{2} + 3 (- 16 x) + 3 \cdot 4$

Step 7.2

Multiply $- 16$ by $3$ .

$g (4 x - 2) = 48 x^{2} - 48 x + 3 \cdot 4$

Step 7.3

Multiply $3$ by $4$ .

$g (4 x - 2) = 48 x^{2} - 48 x + 12$

Step 8

Evaluate the result function by replacing $x$ with $\frac{3}{2}$ in the function.

$48 {(\frac{3}{2})}^{2} - 48 (\frac{3}{2}) + 12$

Step 9

Simplify the evaluated function.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the product rule to $\frac{3}{2}$ .

$48 \frac{3^{2}}{2^{2}} - 48 (\frac{3}{2}) + 12$

Step 9.1.2

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

$48 \frac{9}{2^{2}} - 48 (\frac{3}{2}) + 12$

Step 9.1.3

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

$48 (\frac{9}{4}) - 48 (\frac{3}{2}) + 12$

Step 9.1.4

Cancel the common factor of $4$ .

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Step 9.1.4.1

Factor $4$ out of $48$ .

$4 (12) \frac{9}{4} - 48 (\frac{3}{2}) + 12$

Step 9.1.4.2

Cancel the common factor.

$4 \cdot 12 \frac{9}{4} - 48 (\frac{3}{2}) + 12$

Step 9.1.4.3

Rewrite the expression.

$12 \cdot 9 - 48 (\frac{3}{2}) + 12$

Step 9.1.5

Multiply $12$ by $9$ .

$108 - 48 (\frac{3}{2}) + 12$

Step 9.1.6

Cancel the common factor of $2$ .

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Step 9.1.6.1

Factor $2$ out of $- 48$ .

$108 + 2 (- 24) \frac{3}{2} + 12$

Step 9.1.6.2

Cancel the common factor.

$108 + 2 \cdot - 24 \frac{3}{2} + 12$

Step 9.1.6.3

Rewrite the expression.

$108 - 24 \cdot 3 + 12$

Step 9.1.7

Multiply $- 24$ by $3$ .

$108 - 72 + 12$

Step 9.2

Simplify by adding and subtracting.

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Step 9.2.1

Subtract $72$ from $108$ .

$36 + 12$

Step 9.2.2

Add $36$ and $12$ .

$48$