Question

Please help Show that the inverse of a linear function y=mx+b, where m≠0 and x≠b, is also a linear function

Answer 1

Let f(x) = mx+b. Also, let g(x) be the inverse of f(x)

To find the inverse, we start with y = mx+b and swap x and y. From there, we solve for y like so

y = mx+b

x = my + b

x-b = my

my = x-b

y = (x-b)/m ... note m is in the denominator, so m cannot be 0

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

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

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

This new equation is linear because it is in the form (slope)x+(y intercept)

The new slope is 1/m and the new y intercept is -b/m

So this proves that the inverse g(x) is linear when f(x) is linear.