If f is a linear function such that f(x + 5) - f(x)=6, find the value of inverse f(x + 4) - inverse f(x).

Answer 1

Since f(x) is a linear function, f(x) = ax + b, for some real numbers a and b.

Since f(x+5) - f(x) = 6, we have [a(x+5) + b] - (ax + b) = 6. So, 5a = 6. Therefore, a = 6/5.

So, f(x) = (6/5)x + b

Find f^-1(x):

y = (6/5)x + b

Switch x and y to get x = (6/5)y + b

Solve for y: x - b = (6/5)y

So, y = f^-1(x) = (5x - 5b) / 6

f^-1(x+4) - f^-1(x) = [5(x+4) - 5b] / 6 - (5x - 5b) / 6 = 20/6 = 10/3