Question

composite functions & inverses
perform the indicated operation.

1. $f(x)=3x-2$

$g(x)=x^2-3x$
find $f(g(x))$
find $g(f(x))$
$f= 3(x^2-3x)-2$
$3x^2-9x-2$
$g = x^2(3x-2)-3$
$3x^2-2-3$
$3x^2-5$

1. $g(t)=$

Explanation:

Step1: Substitute $g(x)$ into $f(x)$

$f(g(x)) = 3(x^2 - 3x) - 2$

Step2: Expand and simplify

$f(g(x)) = 3x^2 - 9x - 2$

Step3: Substitute $f(x)$ into $g(x)$

$g(f(x)) = (3x - 2)^2 - 3(3x - 2)$

Step4: Expand the squared term

$(3x - 2)^2 = 9x^2 - 12x + 4$

Step5: Expand the linear term

$-3(3x - 2) = -9x + 6$

Step6: Combine and simplify

$g(f(x)) = 9x^2 - 12x + 4 - 9x + 6 = 9x^2 - 21x + 10$

Answer:

$f(g(x)) = 3x^2 - 9x - 2$
$g(f(x)) = 9x^2 - 21x + 10$