Find f(g(x)) f(x)=x^2-4x+5 , g(x)=x-3

$f (x) = x^{2} - 4 x + 5$ , $g (x) = x - 3$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x - 3)$ by substituting in the value of $g$ into $f$ .

$f (x - 3) = {(x - 3)}^{2} - 4 (x - 3) + 5$

Step 3

Simplify each term.

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

$f (x - 3) = (x - 3) (x - 3) - 4 (x - 3) + 5$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$f (x - 3) = x (x - 3) - 3 (x - 3) - 4 (x - 3) + 5$

Apply the distributive property.

$f (x - 3) = x \cdot x + x \cdot - 3 - 3 (x - 3) - 4 (x - 3) + 5$

Apply the distributive property.

$f (x - 3) = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 4 (x - 3) + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x - 3) = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - 4 (x - 3) + 5$

Move $- 3$ to the left of $x$ .

$f (x - 3) = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - 4 (x - 3) + 5$

Multiply $- 3$ by $- 3$ .

$f (x - 3) = x^{2} - 3 x - 3 x + 9 - 4 (x - 3) + 5$

Subtract $3 x$ from $- 3 x$ .

$f (x - 3) = x^{2} - 6 x + 9 - 4 (x - 3) + 5$

Apply the distributive property.

$f (x - 3) = x^{2} - 6 x + 9 - 4 x - 4 \cdot - 3 + 5$

Multiply $- 4$ by $- 3$ .

$f (x - 3) = x^{2} - 6 x + 9 - 4 x + 12 + 5$

Step 4

Simplify by adding terms.

Subtract $4 x$ from $- 6 x$ .

$f (x - 3) = x^{2} - 10 x + 9 + 12 + 5$

Simplify by adding numbers.

Add $9$ and $12$ .

$f (x - 3) = x^{2} - 10 x + 21 + 5$

Add $21$ and $5$ .

$f (x - 3) = x^{2} - 10 x + 26$