opatovaL

2021-02-24

For $f\left(x\right)=6/x$ and $g\left(x\right)=6/x$, find the following functions.

a) $\left(f\cdot g\right)\left(x\right)$

b) $\left(g\cdot f\right)\left(x\right)$

c) $\left(f\cdot g\right)\left(7\right)$

d) $\left(g\cdot f\right)\left(7\right)$

Leonard Stokes

a) $\left(f\cdot g\right)\left(x\right)=f\left(g\left(x\right)\right)\left({:}^{\prime }\left(f\cdot g\right)\left(X\right)=f\left(g\left(x\right)\right)$

$=f\left(6/x\right)\left({:}^{\prime }g\left(x\right)=6/x\right)$
$=6/\left(6/x\right)\left({:}^{\prime }f\left(x\right)=6/x\right)$

$=\left(6x\right)/6$

$=x$

b) Obtain the composition function g of f as follows.

$\left(g\cdot f\right)\left(x\right)=g\left(f\left(x\right)\right)\left({:}^{\prime }\left(f\cdot g\right)\left(x\right)=f\left(g\left(x\right)\right)$

$=g\left(6/x\right)\left({:}^{\prime }g\left(x\right)=6/x\right)$

$=6/\left(6/x\right)\left({:}^{\prime }g\left(x\right)=6/x\right)$

$=6/6\cdot x=x$

c) Obtain the value of $\left(f\cdot g\right)\left(7\right)$ as follows.
Substitute for x in $\left(f\cdot g\right)\left(x\right)=x$

$\left(f\cdot g\right)\left(7\right)=7$

Thus, the value of $\left(f\cdot g\right)\left(7\right)$ is 7

d) Obtain the value of $\left(g\cdot f\right)\left(7\right)$ as follows.

Substitute for x in $\left(g\cdot f\right)\left(x\right)=x$

$\left(g\cdot f\right)\left(7\right)=7$

Thus, the value of $\left(g\cdot f\right)\left(7\right)$ is 7

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