Find f(g(x)) f(x)=3x^2-4 g(x)=x^2-8x+4

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} - 8 x + 4)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

Rewrite ${(x^{2} - 8 x + 4)}^{2}$ as $(x^{2} - 8 x + 4) (x^{2} - 8 x + 4)$ .

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

Expand $(x^{2} - 8 x + 4) (x^{2} - 8 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

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

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.

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

Add $2$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 8 x^{1 + 2} + x^{2} \cdot 4 - 8 x \cdot x^{2} - 8 x (- 8 x) - 8 x \cdot 4 + 4 x^{2} + 4 (- 8 x) + 4 \cdot 4) - 4$

Add $1$ and $2$ .

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

Move $4$ to the left of $x^{2}$ .

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

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

Move $x^{2}$ .

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

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

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

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

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

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{2 + 1} - 8 x (- 8 x) - 8 x \cdot 4 + 4 x^{2} + 4 (- 8 x) + 4 \cdot 4) - 4$

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 8$ by $- 8$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 64 x^{2} - 8 x \cdot 4 + 4 x^{2} + 4 (- 8 x) + 4 \cdot 4) - 4$

Multiply $4$ by $- 8$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 64 x^{2} - 32 x + 4 x^{2} + 4 (- 8 x) + 4 \cdot 4) - 4$

Multiply $- 8$ by $4$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 64 x^{2} - 32 x + 4 x^{2} - 32 x + 4 \cdot 4) - 4$

Multiply $4$ by $4$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 8 x^{3} + 4 x^{2} - 8 x^{3} + 64 x^{2} - 32 x + 4 x^{2} - 32 x + 16) - 4$

Subtract $8 x^{3}$ from $- 8 x^{3}$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 16 x^{3} + 4 x^{2} + 64 x^{2} - 32 x + 4 x^{2} - 32 x + 16) - 4$

Add $4 x^{2}$ and $64 x^{2}$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 16 x^{3} + 68 x^{2} - 32 x + 4 x^{2} - 32 x + 16) - 4$

Add $68 x^{2}$ and $4 x^{2}$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 16 x^{3} + 72 x^{2} - 32 x - 32 x + 16) - 4$

Subtract $32 x$ from $- 32 x$ .

$f (x^{2} - 8 x + 4) = 3 (x^{4} - 16 x^{3} + 72 x^{2} - 64 x + 16) - 4$

Apply the distributive property.

$f (x^{2} - 8 x + 4) = 3 x^{4} + 3 (- 16 x^{3}) + 3 (72 x^{2}) + 3 (- 64 x) + 3 \cdot 16 - 4$

Simplify.

Multiply $- 16$ by $3$ .

$f (x^{2} - 8 x + 4) = 3 x^{4} - 48 x^{3} + 3 (72 x^{2}) + 3 (- 64 x) + 3 \cdot 16 - 4$

Multiply $72$ by $3$ .

$f (x^{2} - 8 x + 4) = 3 x^{4} - 48 x^{3} + 216 x^{2} + 3 (- 64 x) + 3 \cdot 16 - 4$

Multiply $- 64$ by $3$ .

$f (x^{2} - 8 x + 4) = 3 x^{4} - 48 x^{3} + 216 x^{2} - 192 x + 3 \cdot 16 - 4$

Multiply $3$ by $16$ .

$f (x^{2} - 8 x + 4) = 3 x^{4} - 48 x^{3} + 216 x^{2} - 192 x + 48 - 4$

Step 4

Subtract $4$ from $48$ .

$f (x^{2} - 8 x + 4) = 3 x^{4} - 48 x^{3} + 216 x^{2} - 192 x + 44$

Find f(g(x)) f(x)=3x^2-4 g(x)=x^2-8x+4

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