Find f^-1(x) if f(x)=(x^2+4)?

1 Answer
Steve M
Sep 9, 2017

# f^(-1)(x) = +- sqrt(x-4) #

Explanation:

We have:

# f(x) = x^2+4 #

And we seek the inverse function, #f^(-1)(x)#.

My preferred approach is to put:

# y = x^2+4 # ..... [A]

And rearrange to form an explicit relationship #x=f(y)#, and this function is the inverse, #f^(-1)(x)#

So, from [A] we have:

# x^2 = y-4 #
# :. x = +- sqrt(y-4) #

Thus:

# f^(-1)(x) = +- sqrt(x-4) #

Note that by the formal definition #f^(-1)(x)# is not a function, as it is multi-valued. This occurs because squaring either a positive or negative number in the original function yields a positive number:

# f(-1) = f(1) = 5 => f^(-1)(5) = +- 1#

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