How do you find the compositions given #f(x)=8x# and #g(x)=x/8#?

Question

Theodore Amidon · Accepted Answer

#fog : RR rarr RR,# is defined by, #fog(x)=x.#

#gof : RR rarr RR,# is defined by #gof(x)=x.# We can easily see that the Domain & Range of funs. #f & g# are #RR.# Thus, #f, g :RR rarr RR.# We denote by #R_f# & #D_f# the Range & Domain of fun. #f#, resp. For #fog# may become defined, we must have, #R_g sub D_f,# & similarly, for #gof#, #R_f sub D_g.# Clearly, these conds. are satisfied, we find that both #fog# & #gof# are defined, and, #fog : RR rarr RR, gof : RR rarr RR.# As for formula of #fog#, we have, #fog(x)=f(g(x))=f(u),# say, where #u=g(x)# #=8u#.......[since,#f(x)=8x]# #=8g(x)#.....[since #u=g(x)]# #=8(x/8)#..........[since #g(x)=x/8 #=x# Thus, #fog : RR rarr RR,# is defined by, #fog(x)=x.# Likewise, we can demonstrate that, #gof : RR rarr RR,# is defined by #gof(x)=x.#

Lily Anderson · Answer

undefined To find the compositions of the functions $f(x) = 8x$ and $g(x) = \frac{x}{8}$, we calculate $f(g(x))$ and $g(f(x))$.

$f(g(x)) = f\left(\frac{x}{8}\right) = 8\left(\frac{x}{8}\right) = x$

$g(f(x)) = g(8x) = \frac{8x}{8} = x$

So, both compositions are equal to $x$.