Calculus Examples

$f (x) = x^{2} + 3 x - 1$ , $g (x) = x + 7$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x + 7)$ by substituting in the value of $g$ into $f$ .

$f (x + 7) = {(x + 7)}^{2} + 3 (x + 7) - 1$

Step 3

Simplify each term.

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Step 3.1

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

$f (x + 7) = (x + 7) (x + 7) + 3 (x + 7) - 1$

Step 3.2

Expand $(x + 7) (x + 7)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$f (x + 7) = x (x + 7) + 7 (x + 7) + 3 (x + 7) - 1$

Step 3.2.2

Apply the distributive property.

$f (x + 7) = x \cdot x + x \cdot 7 + 7 (x + 7) + 3 (x + 7) - 1$

Step 3.2.3

Apply the distributive property.

$f (x + 7) = x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7 + 3 (x + 7) - 1$

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 $x$ .

$f (x + 7) = x^{2} + x \cdot 7 + 7 x + 7 \cdot 7 + 3 (x + 7) - 1$

Step 3.3.1.2

Move $7$ to the left of $x$ .

$f (x + 7) = x^{2} + 7 \cdot x + 7 x + 7 \cdot 7 + 3 (x + 7) - 1$

Step 3.3.1.3

Multiply $7$ by $7$ .

$f (x + 7) = x^{2} + 7 x + 7 x + 49 + 3 (x + 7) - 1$

Step 3.3.2

Add $7 x$ and $7 x$ .

$f (x + 7) = x^{2} + 14 x + 49 + 3 (x + 7) - 1$

Step 3.4

Apply the distributive property.

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

Step 3.5

Multiply $3$ by $7$ .

$f (x + 7) = x^{2} + 14 x + 49 + 3 x + 21 - 1$

Step 4

Simplify by adding terms.

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Step 4.1

Add $14 x$ and $3 x$ .

$f (x + 7) = x^{2} + 17 x + 49 + 21 - 1$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $49$ and $21$ .

$f (x + 7) = x^{2} + 17 x + 70 - 1$

Step 4.2.2

Subtract $1$ from $70$ .

$f (x + 7) = x^{2} + 17 x + 69$