Find a linear function 𝑓(𝑥) = 𝑎𝑥 + 𝑏 so that the graphs of f(x) and f^-1(x) are parallel and do not intersect.

Question

Explain your reasoning.

Answer 1

IF y=ax+b, y=f(x)

solve for x

ax+y-b

x = (y-b)/a

now switch the x and y to get

y=(x-b)/a = f^-1(x) the inverse function.

f^-1(x) = x/a - b/a

for this to be parallel to the original f(x) set the slopes equal

a=1/a

a^2 = 1

a = -1 or +1

y=x+b has an inverse function y=x-b

y=-x+b has an inverse function y=-x+b

in each case they are parallel

they have the same slope but different y intercepts

b can be any real number, except zero

for example:

f(x) = x+5 has a parallel inverse function f^-1(x) = x-5

those two functions have graphs which are straight lines

which are mirror images of each other if a mirror were

put through the origin at a 45 degree angle.