# Find the composite functions f @ g and g @ f. Find the domain of each composite function. Are the two composite functions equalf(x) = 3x + 1 g(x) = −x

Find the composite functions $f\circ g$ and $g\circ f$. Find the domain of each composite function. Are the two composite functions equal
$f\left(x\right)=3x+1$
$g\left(x\right)=-x$

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faldduE

Consider the function:
$f\left(x\right)=3x+1, g\left(x\right)=-x$
To calculate $\left(f\circ g\right)$
$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$
$=3\left(g\left(x\right)\right)+1$
$=3\left(-x\right)+1$
$=-3x+1$
To calculate $\left(g\circ f\right)$
$\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$
$=g\left(3x+1\right)$
$=-\left(3x+1\right)$
$=-3x-1$
To calculate the domain of $\left(f\circ g\right)$ and $\left(g\circ f\right)$ function, The domain of a function is the set of input values for which the function is real or defined,
Here,The composite function $\left(f\circ g\right)$ and $\left(g\circ f\right)$ has no undefined points.
Therefore the domain of both composite functions is $-\mathrm{\infty }
Both composite functions are not equal.