How do you verify that f(x) and g(x) are inverses: #f(x) = x+7#, #g(x) = x-7#?

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Answer 1 · 2015-10-21T09:28:39

There are two methods of checking if #f(x)# and #g(x)# are inverse functions. See explanation for details.

Method 1

First method is to look for inverse function of both functions.

Example.

We are looking for inverse function of #f(x)=x+7#

From the expression #y=x+7# we try to calculate #x#

#y=x+7#

#x=y-7#, so we found that #g(x)# is inverse of #f(x)#.

Now we have to look for the inverse of #g(x)#

#g(x)=x-7#

#y=x-7#

#x=y+7#

So we found that #f(x)# is the inverse function of #g(x)#

If #f# is inverse of #g# and #g# is inverse of #f# then #f# and #g# are inverse functions.

Method 2

The second way is to find the compound functions #f(g(x))# and #g(f(x))#. If they both are #h(x)=x# then #f# and #g# are inverse.

Example:

#f(g(x)=[x-7]+7# The expression in brackets is #g(x)# inserted as #x#

#f(g(x))=x-7+7=x#

#g(f(x)=[x+7]-7# The expression in brackets is #f(x)# inserted as #x#

#g(f(x))=x+7-7=x#

We found out that: #f(g(x))=g(f(x))=x#. This concludes the proof, that #f(x)# and #g(x)# are inverse functions.