Find f(g(x)) f(x)=49-x^2 , g(x)=7-x

$f (x) = 49 - x^{2}$ , $g (x) = 7 - x$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (7 - x)$ by substituting in the value of $g$ into $f$ .

$f (7 - x) = 49 - {(7 - x)}^{2}$

Step 3

Simplify each term.

Rewrite ${(7 - x)}^{2}$ as $(7 - x) (7 - x)$ .

$f (7 - x) = 49 - ((7 - x) (7 - x))$

Expand $(7 - x) (7 - x)$ using the FOIL Method.

Apply the distributive property.

$f (7 - x) = 49 - (7 (7 - x) - x (7 - x))$

Apply the distributive property.

$f (7 - x) = 49 - (7 \cdot 7 + 7 (- x) - x (7 - x))$

Apply the distributive property.

$f (7 - x) = 49 - (7 \cdot 7 + 7 (- x) - x \cdot 7 - x (- x))$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$f (7 - x) = 49 - (49 + 7 (- x) - x \cdot 7 - x (- x))$

Multiply $- 1$ by $7$ .

$f (7 - x) = 49 - (49 - 7 x - x \cdot 7 - x (- x))$

Multiply $7$ by $- 1$ .

$f (7 - x) = 49 - (49 - 7 x - 7 x - x (- x))$

Rewrite using the commutative property of multiplication.

$f (7 - x) = 49 - (49 - 7 x - 7 x - 1 \cdot (- 1 x \cdot x))$

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

Move $x$ .

$f (7 - x) = 49 - (49 - 7 x - 7 x - 1 \cdot (- 1 (x \cdot x)))$

Multiply $x$ by $x$ .

$f (7 - x) = 49 - (49 - 7 x - 7 x - 1 \cdot (- 1 x^{2}))$

Multiply $- 1$ by $- 1$ .

$f (7 - x) = 49 - (49 - 7 x - 7 x + 1 x^{2})$

Multiply $x^{2}$ by $1$ .

$f (7 - x) = 49 - (49 - 7 x - 7 x + x^{2})$

Subtract $7 x$ from $- 7 x$ .

$f (7 - x) = 49 - (49 - 14 x + x^{2})$

Apply the distributive property.

$f (7 - x) = 49 - 1 \cdot 49 - (- 14 x) - x^{2}$

Simplify.

Multiply $- 1$ by $49$ .

$f (7 - x) = 49 - 49 - (- 14 x) - x^{2}$

Multiply $- 14$ by $- 1$ .

$f (7 - x) = 49 - 49 + 14 x - x^{2}$

Step 4

Simplify the expression.

Subtract $49$ from $49$ .

$f (7 - x) = 0 + 14 x - x^{2}$

Add $0$ and $14 x$ .

$f (7 - x) = 14 x - x^{2}$

Reorder $14 x$ and $- x^{2}$ .

$f (7 - x) = - x^{2} + 14 x$

Find f(g(x)) f(x)=49-x^2 , g(x)=7-x

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