How do you find the inverse of #y = - |x-3| + 5#?

1 Answer
Feb 9, 2017

Answer:

#f^(-1)(x) = 3 pm (5 - x)#

Explanation:

#y = -|x-3| + 5 #

# |x-3| = 5 - y#

# x-3 = pm (5 - y)#

# x = 3 pm (5 - y)#

The inverse function is therefore:

#f^(-1)(x) = 3 pm (5 - x)#

There would be no harm in verifying this for #x = 2# and #x = 4#, ie either side of #x = 3# which is the point at which #(x-3)# changes sign.

For #x = color(red)(2)#, we get #y = 4#. Our inverse function suggests that #f^(-1)(4) = 3 pm 1 = 4, color(red)(2)#.

And for #x = color(blue)(4)#, we also get #y = 4#, which we expect from the symmetry. Our inverse function suggests that #f^(-1)(4) = 3 pm 1 = color(blue)(4),2#.

graph{y = -|x-3| + 5 [-10, 10, -5, 5]}