Question #8ed74

1 Answer
Sep 17, 2017

#f^(-1) (x) = sqrt(-x + 10)#

Explanation:

#f(x) = 10 - x^2, x >= 0 #

To find the inverse, #f^(-1) (x)#, switch the variables #x# and #y# and solve for #y#.

#y = 10 - x^2#

#x = 10 - y^2# #-># switch variables

# x- 10 = - y^2#

# -x + 10 = y^2#

# y = +- sqrt(-x + 10)#

However, since the domain of #f(x)# is restricted to #x>=0# (only positive values of #x#), the range of #f^(-1)(x)# will be restricted to only positive values of #y#. Thus, we only take the positive square root above.

So, #f^(-1) (x) = sqrt(-x + 10)#.

Any function and its inverse will be reflections of each other over the line #y = x#. Graphing #f(x)# and #f^(-1)(x)#, we can verify that they are indeed inverses.

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