Find f(h(x)) f(x)=5x^2+2x-5 , g(x)=7x+8 , h(x)=4x^2-10

$f (x) = 5 x^{2} + 2 x - 5$ , $g (x) = 7 x + 8$ , $h (x) = 4 x^{2} - 10$

Step 1

Set up the composite result function.

$f (h (x))$

Step 2

Evaluate $f (4 x^{2} - 10)$ by substituting in the value of $h$ into $f$ .

$f (4 x^{2} - 10) = 5 {(4 x^{2} - 10)}^{2} + 2 (4 x^{2} - 10) - 5$

Step 3

Simplify each term.

Rewrite ${(4 x^{2} - 10)}^{2}$ as $(4 x^{2} - 10) (4 x^{2} - 10)$ .

$f (4 x^{2} - 10) = 5 ((4 x^{2} - 10) (4 x^{2} - 10)) + 2 (4 x^{2} - 10) - 5$

Expand $(4 x^{2} - 10) (4 x^{2} - 10)$ using the FOIL Method.

Apply the distributive property.

$f (4 x^{2} - 10) = 5 (4 x^{2} (4 x^{2} - 10) - 10 (4 x^{2} - 10)) + 2 (4 x^{2} - 10) - 5$

Apply the distributive property.

$f (4 x^{2} - 10) = 5 (4 x^{2} (4 x^{2}) + 4 x^{2} \cdot - 10 - 10 (4 x^{2} - 10)) + 2 (4 x^{2} - 10) - 5$

Apply the distributive property.

$f (4 x^{2} - 10) = 5 (4 x^{2} (4 x^{2}) + 4 x^{2} \cdot - 10 - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (4 x^{2} - 10) = 5 (4 \cdot (4 x^{2} x^{2}) + 4 x^{2} \cdot - 10 - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

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

Move $x^{2}$ .

$f (4 x^{2} - 10) = 5 (4 \cdot (4 (x^{2} x^{2})) + 4 x^{2} \cdot - 10 - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

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

$f (4 x^{2} - 10) = 5 (4 \cdot (4 x^{2 + 2}) + 4 x^{2} \cdot - 10 - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

Add $2$ and $2$ .

$f (4 x^{2} - 10) = 5 (4 \cdot (4 x^{4}) + 4 x^{2} \cdot - 10 - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

Multiply $4$ by $4$ .

$f (4 x^{2} - 10) = 5 (16 x^{4} + 4 x^{2} \cdot - 10 - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

Multiply $- 10$ by $4$ .

$f (4 x^{2} - 10) = 5 (16 x^{4} - 40 x^{2} - 10 (4 x^{2}) - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

Multiply $4$ by $- 10$ .

$f (4 x^{2} - 10) = 5 (16 x^{4} - 40 x^{2} - 40 x^{2} - 10 \cdot - 10) + 2 (4 x^{2} - 10) - 5$

Multiply $- 10$ by $- 10$ .

$f (4 x^{2} - 10) = 5 (16 x^{4} - 40 x^{2} - 40 x^{2} + 100) + 2 (4 x^{2} - 10) - 5$

Subtract $40 x^{2}$ from $- 40 x^{2}$ .

$f (4 x^{2} - 10) = 5 (16 x^{4} - 80 x^{2} + 100) + 2 (4 x^{2} - 10) - 5$

Apply the distributive property.

$f (4 x^{2} - 10) = 5 (16 x^{4}) + 5 (- 80 x^{2}) + 5 \cdot 100 + 2 (4 x^{2} - 10) - 5$

Simplify.

Multiply $16$ by $5$ .

$f (4 x^{2} - 10) = 80 x^{4} + 5 (- 80 x^{2}) + 5 \cdot 100 + 2 (4 x^{2} - 10) - 5$

Multiply $- 80$ by $5$ .

$f (4 x^{2} - 10) = 80 x^{4} - 400 x^{2} + 5 \cdot 100 + 2 (4 x^{2} - 10) - 5$

Multiply $5$ by $100$ .

$f (4 x^{2} - 10) = 80 x^{4} - 400 x^{2} + 500 + 2 (4 x^{2} - 10) - 5$

Apply the distributive property.

$f (4 x^{2} - 10) = 80 x^{4} - 400 x^{2} + 500 + 2 (4 x^{2}) + 2 \cdot - 10 - 5$

Multiply $4$ by $2$ .

$f (4 x^{2} - 10) = 80 x^{4} - 400 x^{2} + 500 + 8 x^{2} + 2 \cdot - 10 - 5$

Multiply $2$ by $- 10$ .

$f (4 x^{2} - 10) = 80 x^{4} - 400 x^{2} + 500 + 8 x^{2} - 20 - 5$

Step 4

Simplify by adding terms.

Add $- 400 x^{2}$ and $8 x^{2}$ .

$f (4 x^{2} - 10) = 80 x^{4} - 392 x^{2} + 500 - 20 - 5$

Simplify by subtracting numbers.

Subtract $20$ from $500$ .

$f (4 x^{2} - 10) = 80 x^{4} - 392 x^{2} + 480 - 5$

Subtract $5$ from $480$ .

$f (4 x^{2} - 10) = 80 x^{4} - 392 x^{2} + 475$

Find f(h(x)) f(x)=5x^2+2x-5 , g(x)=7x+8 , h(x)=4x^2-10

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