Find f(g(x)) f(x)=x^3-7x^2+2x+11 , g(x)=x+1

$f (x) = x^{3} - 7 x^{2} + 2 x + 11$ , $g (x) = x + 1$

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 $f$ .

$f (x + 1) = {(x + 1)}^{3} - 7 {(x + 1)}^{2} + 2 (x + 1) + 11$

Step 3

Simplify each term.

Use the Binomial Theorem.

$f (x + 1) = x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3} - 7 {(x + 1)}^{2} + 2 (x + 1) + 11$

Simplify each term.

Multiply $3$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3} - 7 {(x + 1)}^{2} + 2 (x + 1) + 11$

One to any power is one.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3} - 7 {(x + 1)}^{2} + 2 (x + 1) + 11$

Multiply $3$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1^{3} - 7 {(x + 1)}^{2} + 2 (x + 1) + 11$

One to any power is one.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 {(x + 1)}^{2} + 2 (x + 1) + 11$

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

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 ((x + 1) (x + 1)) + 2 (x + 1) + 11$

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

Apply the distributive property.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x (x + 1) + 1 (x + 1)) + 2 (x + 1) + 11$

Apply the distributive property.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x \cdot x + x \cdot 1 + 1 (x + 1)) + 2 (x + 1) + 11$

Apply the distributive property.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 2 (x + 1) + 11$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 2 (x + 1) + 11$

Multiply $x$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x^{2} + x + 1 x + 1 \cdot 1) + 2 (x + 1) + 11$

Multiply $x$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x^{2} + x + x + 1 \cdot 1) + 2 (x + 1) + 11$

Multiply $1$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x^{2} + x + x + 1) + 2 (x + 1) + 11$

Add $x$ and $x$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 (x^{2} + 2 x + 1) + 2 (x + 1) + 11$

Apply the distributive property.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 x^{2} - 7 (2 x) - 7 \cdot 1 + 2 (x + 1) + 11$

Simplify.

Multiply $2$ by $- 7$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 x^{2} - 14 x - 7 \cdot 1 + 2 (x + 1) + 11$

Multiply $- 7$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 x^{2} - 14 x - 7 + 2 (x + 1) + 11$

Apply the distributive property.

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 x^{2} - 14 x - 7 + 2 x + 2 \cdot 1 + 11$

Multiply $2$ by $1$ .

$f (x + 1) = x^{3} + 3 x^{2} + 3 x + 1 - 7 x^{2} - 14 x - 7 + 2 x + 2 + 11$

Step 4

Simplify by adding terms.

Subtract $7 x^{2}$ from $3 x^{2}$ .

$f (x + 1) = x^{3} - 4 x^{2} + 3 x + 1 - 14 x - 7 + 2 x + 2 + 11$

Subtract $14 x$ from $3 x$ .

$f (x + 1) = x^{3} - 4 x^{2} - 11 x + 1 - 7 + 2 x + 2 + 11$

Add $- 11 x$ and $2 x$ .

$f (x + 1) = x^{3} - 4 x^{2} - 9 x + 1 - 7 + 2 + 11$

Simplify by adding and subtracting.

Subtract $7$ from $1$ .

$f (x + 1) = x^{3} - 4 x^{2} - 9 x - 6 + 2 + 11$

Add $- 6$ and $2$ .

$f (x + 1) = x^{3} - 4 x^{2} - 9 x - 4 + 11$

Add $- 4$ and $11$ .

$f (x + 1) = x^{3} - 4 x^{2} - 9 x + 7$

Find f(g(x)) f(x)=x^3-7x^2+2x+11 , g(x)=x+1

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