How do you find the inverse of #f(x) = x – 7# and is it a function?

Question

How do you find the inverse of  #f(x) = x – 7# and is it a function?

Chloe Alexander · Accepted Answer

The inverse function is #f^-1(x)=x+7# To find the inverse function of #y=f(x)# you have to calculate #x# as function of #y# and check if the formulka you get describes a function i. e. it has only one possible value for any argument. In this example our function is #y=x-7#. #y=x-7# #y-x=-7# #-x=-y-7# #x=y+7# It is a function. For any number #x# there is only one number #7# greater than #x#, so there is only one value. Now we can write the inverse function using standard notation (##x# for argument and #y# for value). The inverse is #y=x+7#

Lincoln Anderson · Answer

undefined To find the inverse of a function $ f(x) $, you swap the roles of $ x $ and $ y $, then solve for $ y $.

Given $ f(x) = x - 7 $, let $ y = f(x) $, then $ x = y - 7 $.

To find the inverse, solve for $ y $:
$$ x = y - 7 $$
$$ x + 7 = y $$

So, the inverse function $ f^{-1}(x) $ is $ f^{-1}(x) = x + 7 $.

To determine if it is a function, we need to check if each input in the original function corresponds to exactly one output, and vice versa.

Since every value of $ x $ corresponds to exactly one value of $ y $ in both $ f(x) $ and $ f^{-1}(x) $, and every value of $ y $ corresponds to exactly one value of $ x $ in both functions, then both $ f(x) $ and $ f^{-1}(x) $ are functions.

Therefore, $ f^{-1}(x) = x + 7 $ is a function.