How do you show that #f(x)=sqrt(x-4)# and #g(x)=x^2+4# are inverse functions algebraically and graphically?

1 Answer
Noah G
Jan 19, 2017

You can always verify that two functions are inverse functions algebraically by finding their compositions. These should equal #x#.

#f(g(x)) = sqrt(x^2 + 4 - 4)#

#f(g(x)) = sqrt(x^2)#

#f(g(x)) = x#

Try the other composition.

#g(f(x)) = (sqrt(x - 4))^2 + 4#

#g(f(x)) = x - 4 + 4#

#g(f(x)) = x#

These are inverses. Graphically, you can graph both functions an you'll notice that #f(x)# is #g(x)# reflected over the line #y = x# and vice versa. This means that if the point #(1, 2)# lies on a function, then the point #(2, 1)# will lie on its inverse.

Hopefully this helps!

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