Inverse Function.

Question

Let f be a linear function such that f(x+1)-f(x-4)=12,

a) Find the slope of inverse of f(x) .

b) Find the value of f^-1(x+4)-f^(x-2).

Answer 1

If f(x) is a linear function then it can be written as y = mx + b.

f(x + 1) = m(x+1) + b = mx + m +b

f(x + 4) = m(x+4) + b = mx + 4m + b

So f(x+ 1) - f(x+4) = mx + m + b - (mx + 4m +b) = mx + m + b - mx -4m - b = -3m

But we know that value is 12.

So -3m = 12

m = -4

To find the inverse:

y = -4x + b

x = -4y + b

4y = -x + b

y = (-1/4)x + b/4

Call the inverse function g(x).

g(x+1) = (-1/4)(x + 1) + b/4 = (-1/4)x - 1/4 + b/4

g(x+4) = (-1/4)(x+4) + b/4 = (-1/4)x - 1+ b/4

g(x+1) - g(x+4) = (-1/4)x - 1/4 + b/4 - [(-1/4)x - 1 + b/4)] = (-1/4)x - 1/4 + b/4 + (1/4)x + 1 - b/4 = 3/4

These values remain the same regardless of the value of b.

Here is a Desmos graph showing the original function f(x) and its inverse g(x). Use the slider on b to see that the values do not change. Use the slider on x₁ to see the segment joining a point of f with its image on g. Note that the midpoint of that segment lies on the line y = x.

desmos.com/calculator/inda8whaou