Pre calc HELP PLEASE!!!!!!!!!!

Question

Let f(x) = mx + b be the family of all linear functions. Find all values of m and b such that f = f −1 . That is, find all linear functions that are their own inverses.

Answer 1

y=x and y=-x

For linear function y = mx + b

the inverse is x = my + b

(x-b)/m = y

y=(1/m) x - (b/m)

1/m = m and (-b)/m = b

The first equation implies that m=1 or m=-1

for m=-1, (-b)/(-1) = b works for any b.

so y = -x + b is it's own inverse

as shown below.

for m=1, (-b)1 = b

-b = b

b = 0

so y = x is it's own inverse

Proof that y = -x + b is it's own inverse:

y = -x + b

x = -y + b

x - b = -y

-x + b = y

Note that -(-x+b)+b = x - b + b = x for f comp g and g comp f