# Find and simplify in expression for the idicated composite functions. State the domain using interval notation. f(x)=3x-1 g(x)=1/(x+3) Find (g@f)(x)

Find and simplify in expression for the idicated composite functions. State the domain using interval notation.
$f\left(x\right)=3x-1$
$g\left(x\right)=\frac{1}{x+3}$
Find $\left(g\circ f\right)\left(x\right)$
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insonsipthinye
Obtain the composite function, $g\circ f\left(x\right)$
$g\circ f\left(X\right)=g\left(f\left(x\right)\right)$
$=\frac{1}{\left(f\left(x\right)\right)+3}$
$=\frac{1}{\left(3x-1\right)+3}$
$=\frac{1}{3x+2},x\ne -\frac{2}{3}$
Here, the simplified expression for the composite function is
$g\circ f\left(x\right)=\frac{1}{3x+2},x\ne -\frac{2}{3}$
The composite function defined for all values of x except at $x=-\frac{2}{3}$ that is defined all values on the left and right of $x=-\frac{2}{3}$ but not at $x=-\frac{2}{3}$, thus the domain in the interval notation becomes $\left(-\mathrm{\infty },-\frac{2}{3}\right)\cup \left(\frac{2}{3},\mathrm{\infty }\right)$