Verify f(x) 5-2x and g(x) = 5-x/2 are inverse functions

Answer 1

There are a couple different ways to check if one function is the inverse of another. I'll just use one.

The composition of 2 functions will be x if the functions are inverses of each other. We need both compositions to work. Let's try it and see:

f(x) = 5 - 2x

g(x) = 5 - x/2)

f(g(x)) = 5 - 2(5 - x/2)

= 5 - 10 + 2x/2

= -5 + x

Thus, f(x) and g(x) would not be inverses of each other. We do not need to try g(f(x)) because f(g(x)) failed the test.

However, if g(x) = (5-x)/2, then

f(g(x)) = 5 - 2((5 - x)/2)

= 5 - 2(5-x)/2

= 5 - (2/2)(5 - x)

= 5 - (5 - x)

= 0 - (-x)

= x

Now check the other composition: g(f(x))

g(f(x)) = (5 - (5 - 2x))/2

= (5 - 5 + 2x)/2

= (0 + 2x)/2

= 2x/2

= x

Since both g(f(x)) and f(g(x)) = x, then f(x) and g(x) are inverses.