What is the inverse function of #f(x)=sqrt(8-x)#?

2 Answers
Jul 5, 2015

The inverse of #f(x) = sqrt(8-x)# is
#color(white)("XXXX")##g(x)=8-x^2#

Explanation:

Rename #f(x)# as #y#
Then, using the original equation, solve for #x#
#color(white)("XXXX")##y = sqrt(8-x)#

#color(white)("XXXX")##y^2= 8-x#

#color(white)("XXXX")##y^2-8 = -x#

#color(white)("XXXX")##x = 8-y^2#

Rename #x# as #g(y)#
#color(white)("XXXX")##g(y) = 8-y^2#

...or using the more common variable #x# as the function variable:
#color(white)("XXXX")##g(x) = 8-x^2#

Jul 5, 2015

#f^-1(y) = 8-y^2# with domain #[0, oo)# and range #(-oo,8]#

Explanation:

Let #y = f(x) = sqrt(8-x)#

The domain of #f# is #(-oo,8]# and its range is #[0,oo)#

Then #y^2 = 8-x#

Add #x-y^2# to both sides to get:

#x = 8 - y^2#

So #f^-1(y) = 8-y^2# with domain #[0,oo)# and range #(-oo,8]#