Find f(g(x)) f(x)=5x^2-4x , g(x)=5x+1

$f (x) = 5 x^{2} - 4 x$ , $g (x) = 5 x + 1$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (5 x + 1)$ by substituting in the value of $g$ into $f$ .

$f (5 x + 1) = 5 {(5 x + 1)}^{2} - 4 (5 x + 1)$

Step 3

Simplify each term.

Rewrite ${(5 x + 1)}^{2}$ as $(5 x + 1) (5 x + 1)$ .

$f (5 x + 1) = 5 ((5 x + 1) (5 x + 1)) - 4 (5 x + 1)$

Expand $(5 x + 1) (5 x + 1)$ using the FOIL Method.

Apply the distributive property.

$f (5 x + 1) = 5 (5 x (5 x + 1) + 1 (5 x + 1)) - 4 (5 x + 1)$

Apply the distributive property.

$f (5 x + 1) = 5 (5 x (5 x) + 5 x \cdot 1 + 1 (5 x + 1)) - 4 (5 x + 1)$

Apply the distributive property.

$f (5 x + 1) = 5 (5 x (5 x) + 5 x \cdot 1 + 1 (5 x) + 1 \cdot 1) - 4 (5 x + 1)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (5 x + 1) = 5 (5 \cdot (5 x \cdot x) + 5 x \cdot 1 + 1 (5 x) + 1 \cdot 1) - 4 (5 x + 1)$

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

Move $x$ .

$f (5 x + 1) = 5 (5 \cdot (5 (x \cdot x)) + 5 x \cdot 1 + 1 (5 x) + 1 \cdot 1) - 4 (5 x + 1)$

Multiply $x$ by $x$ .

$f (5 x + 1) = 5 (5 \cdot (5 x^{2}) + 5 x \cdot 1 + 1 (5 x) + 1 \cdot 1) - 4 (5 x + 1)$

Multiply $5$ by $5$ .

$f (5 x + 1) = 5 (25 x^{2} + 5 x \cdot 1 + 1 (5 x) + 1 \cdot 1) - 4 (5 x + 1)$

Multiply $5$ by $1$ .

$f (5 x + 1) = 5 (25 x^{2} + 5 x + 1 (5 x) + 1 \cdot 1) - 4 (5 x + 1)$

Multiply $5 x$ by $1$ .

$f (5 x + 1) = 5 (25 x^{2} + 5 x + 5 x + 1 \cdot 1) - 4 (5 x + 1)$

Multiply $1$ by $1$ .

$f (5 x + 1) = 5 (25 x^{2} + 5 x + 5 x + 1) - 4 (5 x + 1)$

Add $5 x$ and $5 x$ .

$f (5 x + 1) = 5 (25 x^{2} + 10 x + 1) - 4 (5 x + 1)$

Apply the distributive property.

$f (5 x + 1) = 5 (25 x^{2}) + 5 (10 x) + 5 \cdot 1 - 4 (5 x + 1)$

Simplify.

Multiply $25$ by $5$ .

$f (5 x + 1) = 125 x^{2} + 5 (10 x) + 5 \cdot 1 - 4 (5 x + 1)$

Multiply $10$ by $5$ .

$f (5 x + 1) = 125 x^{2} + 50 x + 5 \cdot 1 - 4 (5 x + 1)$

Multiply $5$ by $1$ .

$f (5 x + 1) = 125 x^{2} + 50 x + 5 - 4 (5 x + 1)$

Apply the distributive property.

$f (5 x + 1) = 125 x^{2} + 50 x + 5 - 4 (5 x) - 4 \cdot 1$

Multiply $5$ by $- 4$ .

$f (5 x + 1) = 125 x^{2} + 50 x + 5 - 20 x - 4 \cdot 1$

Multiply $- 4$ by $1$ .

$f (5 x + 1) = 125 x^{2} + 50 x + 5 - 20 x - 4$

Step 4

Simplify by adding terms.

Subtract $20 x$ from $50 x$ .

$f (5 x + 1) = 125 x^{2} + 30 x + 5 - 4$

Subtract $4$ from $5$ .

$f (5 x + 1) = 125 x^{2} + 30 x + 1$

Find f(g(x)) f(x)=5x^2-4x , g(x)=5x+1

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