Find g(f(x)) f(x)=(x-3)/10 , g(x)=10x+3(g*f)(x)

$f (x) = \frac{x - 3}{10}$ , $g (x) = 10 x + 3 (g \cdot f) (x)$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{x - 3}{10})$ by substituting in the value of $f$ into $g$ .

$g (\frac{x - 3}{10}) = 10 (\frac{x - 3}{10}) + 3 (g \cdot f) (\frac{x - 3}{10})$

Step 3

Simplify each term.

Cancel the common factor of $10$ .

Cancel the common factor.

$g (\frac{x - 3}{10}) = 10 (\frac{x - 3}{10}) + 3 (g \cdot f) (\frac{x - 3}{10})$

Rewrite the expression.

$g (\frac{x - 3}{10}) = x - 3 + 3 (g \cdot f) (\frac{x - 3}{10})$

Multiply $3 g f \frac{x - 3}{10}$ .

Combine $3$ and $\frac{x - 3}{10}$ .

$g (\frac{x - 3}{10}) = x - 3 + g f (\frac{3 (x - 3)}{10})$

Combine $g$ and $\frac{3 (x - 3)}{10}$ .

$g (\frac{x - 3}{10}) = x - 3 + f (\frac{g (3 (x - 3))}{10})$

Combine $f$ and $\frac{g (3 (x - 3))}{10}$ .

$g (\frac{x - 3}{10}) = x - 3 + \frac{f (g (3 (x - 3)))}{10}$

Move $3$ to the left of $f g$ .

$g (\frac{x - 3}{10}) = x - 3 + \frac{3 f g (x - 3)}{10}$

Step 4

To write $x$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$g (\frac{x - 3}{10}) = x \cdot \frac{10}{10} + \frac{3 f g (x - 3)}{10} - 3$

Step 5

Simplify terms.

Combine $x$ and $\frac{10}{10}$ .

$g (\frac{x - 3}{10}) = \frac{x \cdot 10}{10} + \frac{3 f g (x - 3)}{10} - 3$

Combine the numerators over the common denominator.

$g (\frac{x - 3}{10}) = \frac{x \cdot 10 + 3 f g (x - 3)}{10} - 3$

Step 6

Simplify the numerator.

Move $10$ to the left of $x$ .

$g (\frac{x - 3}{10}) = \frac{10 \cdot x + 3 f g (x - 3)}{10} - 3$

Apply the distributive property.

$g (\frac{x - 3}{10}) = \frac{10 x + 3 f g x + 3 f g \cdot - 3}{10} - 3$

Multiply $- 3$ by $3$ .

$g (\frac{x - 3}{10}) = \frac{10 x + 3 f g x - 9 f g}{10} - 3$

Step 7

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$g (\frac{x - 3}{10}) = \frac{10 x + 3 f g x - 9 f g}{10} - 3 \cdot \frac{10}{10}$

Step 8

Simplify terms.

Combine $- 3$ and $\frac{10}{10}$ .

$g (\frac{x - 3}{10}) = \frac{10 x + 3 f g x - 9 f g}{10} + \frac{- 3 \cdot 10}{10}$

Combine the numerators over the common denominator.

$g (\frac{x - 3}{10}) = \frac{10 x + 3 f g x - 9 f g - 3 \cdot 10}{10}$

Step 9

Simplify the numerator.

Multiply $- 3$ by $10$ .

$g (\frac{x - 3}{10}) = \frac{10 x + 3 f g x - 9 f g - 30}{10}$

Rewrite $10 x + 3 f g x - 9 f g - 30$ in a factored form.

Reorder terms.

$g (\frac{x - 3}{10}) = \frac{3 f g x - 9 f g + 10 x - 30}{10}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$g (\frac{x - 3}{10}) = \frac{(3 f g x - 9 f g) + 10 x - 30}{10}$

Factor out the greatest common factor (GCF) from each group.

$g (\frac{x - 3}{10}) = \frac{3 f g (x - 3) + 10 (x - 3)}{10}$

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$g (\frac{x - 3}{10}) = \frac{(x - 3) (3 f g + 10)}{10}$

Find g(f(x)) f(x)=(x-3)/10 , g(x)=10x+3(g*f)(x)

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