Find f(f(x)) f(x)=x^2-2x+3 , f(x)=-2x+19

$f (x) = x^{2} - 2 x + 3$ , $f (x) = - 2 x + 19$

Step 1

Set up the composite result function.

$f (f (f (f (x))))$

Step 2

Evaluate $f (x^{2} - 2 x + 3)$ by substituting in the value of $f$ into $f$ .

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

Step 3

Simplify each term.

Rewrite ${(x^{2} - 2 x + 3)}^{2}$ as $(x^{2} - 2 x + 3) (x^{2} - 2 x + 3)$ .

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

Expand $(x^{2} - 2 x + 3) (x^{2} - 2 x + 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

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

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

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

Add $2$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Move $3$ to the left of $x^{2}$ .

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $3$ by $- 2$ .

$f (x^{2} - 2 x + 3) = x^{4} - 2 x^{3} + 3 x^{2} - 2 x^{3} + 4 x^{2} - 6 x + 3 x^{2} + 3 (- 2 x) + 3 \cdot 3 - 2 (x^{2} - 2 x + 3) + 3$

Multiply $- 2$ by $3$ .

$f (x^{2} - 2 x + 3) = x^{4} - 2 x^{3} + 3 x^{2} - 2 x^{3} + 4 x^{2} - 6 x + 3 x^{2} - 6 x + 3 \cdot 3 - 2 (x^{2} - 2 x + 3) + 3$

Multiply $3$ by $3$ .

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

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

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

Add $3 x^{2}$ and $4 x^{2}$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 7 x^{2} - 6 x + 3 x^{2} - 6 x + 9 - 2 (x^{2} - 2 x + 3) + 3$

Add $7 x^{2}$ and $3 x^{2}$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 10 x^{2} - 6 x - 6 x + 9 - 2 (x^{2} - 2 x + 3) + 3$

Subtract $6 x$ from $- 6 x$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 - 2 (x^{2} - 2 x + 3) + 3$

Apply the distributive property.

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 - 2 x^{2} - 2 (- 2 x) - 2 \cdot 3 + 3$

Simplify.

Multiply $- 2$ by $- 2$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 - 2 x^{2} + 4 x - 2 \cdot 3 + 3$

Multiply $- 2$ by $3$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 10 x^{2} - 12 x + 9 - 2 x^{2} + 4 x - 6 + 3$

Step 4

Simplify by adding terms.

Subtract $2 x^{2}$ from $10 x^{2}$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 8 x^{2} - 12 x + 9 + 4 x - 6 + 3$

Add $- 12 x$ and $4 x$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 8 x^{2} - 8 x + 9 - 6 + 3$

Simplify by adding and subtracting.

Subtract $6$ from $9$ .

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

Add $3$ and $3$ .

$f (x^{2} - 2 x + 3) = x^{4} - 4 x^{3} + 8 x^{2} - 8 x + 6$

Find f(f(x)) f(x)=x^2-2x+3 , f(x)=-2x+19

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