Calculus Examples

$f (x) = x^{2} + 3 x + 4$ , $g (x) = x - 1$ , $(f \circ g)$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (x - 1) = {(x - 1)}^{2} + 3 (x - 1) + 4$

Step 3

Simplify each term.

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Step 3.1

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

$f (x - 1) = (x - 1) (x - 1) + 3 (x - 1) + 4$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$f (x - 1) = x (x - 1) - 1 (x - 1) + 3 (x - 1) + 4$

Step 3.2.2

Apply the distributive property.

$f (x - 1) = x \cdot x + x \cdot - 1 - 1 (x - 1) + 3 (x - 1) + 4$

Step 3.2.3

Apply the distributive property.

$f (x - 1) = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

$f (x - 1) = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Step 3.3.1.2

Move $- 1$ to the left of $x$ .

$f (x - 1) = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Step 3.3.1.3

Rewrite $- 1 x$ as $- x$ .

$f (x - 1) = x^{2} - x - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Step 3.3.1.4

Rewrite $- 1 x$ as $- x$ .

$f (x - 1) = x^{2} - x - x - 1 \cdot - 1 + 3 (x - 1) + 4$

Step 3.3.1.5

Multiply $- 1$ by $- 1$ .

$f (x - 1) = x^{2} - x - x + 1 + 3 (x - 1) + 4$

Step 3.3.2

Subtract $x$ from $- x$ .

$f (x - 1) = x^{2} - 2 x + 1 + 3 (x - 1) + 4$

Step 3.4

Apply the distributive property.

$f (x - 1) = x^{2} - 2 x + 1 + 3 x + 3 \cdot - 1 + 4$

Step 3.5

Multiply $3$ by $- 1$ .

$f (x - 1) = x^{2} - 2 x + 1 + 3 x - 3 + 4$

Step 4

Simplify by adding terms.

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Step 4.1

Add $- 2 x$ and $3 x$ .

$f (x - 1) = x^{2} + x + 1 - 3 + 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $3$ from $1$ .

$f (x - 1) = x^{2} + x - 2 + 4$

Step 4.2.2

Add $- 2$ and $4$ .

$f (x - 1) = x^{2} + x + 2$

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