Question

For f(x)=2/3(x-4), determine each of the following:
a) f(5)-f(2)
b)f^-1(x)
c) f(12)-f^-1(4)

Answer 1

a) To find f(5), we plug x = 5 into the function: f(5) = 2/3(5-4) = 2/3(1) = 2/3.
To find f(2), we plug x = 2 into the function: f(2) = 2/3(2-4) = 2/3(-2) = -4/3.
Thus, f(5) - f(2) = 2/3 - (-4/3) = 2/3 + 4/3 = 6/3 = 2.

b) To find f^-1(x), we need to solve for x. Start with the equation y = 2/3(x-4) and solve for x:
Multiply both sides by 3/2 to eliminate the fraction: 3/2(y) = 3/2(2/3)(x-4) = (2/2) (x-4) = x - 4.
Add 4 to both sides to solve for x: x = 3/2(y) + 4.

Thus, the inverse function f^-1(x) = 3/2(x) + 4.

c) To find f(12), we plug x = 12 into the function: f(12) = 2/3(12-4) = 2/3(8) = 16/3.

To find f^-1(4), we need to find the input value that gives an output of 4 when plugged into the inverse function.
Set f^-1(x) = 4 and solve for x:
4 = 3/2(x) + 4
Subtract 4 from both sides: 0 = 3/2(x)
Multiply both sides by 2/3: 0 = x

Thus, f(12) - f^-1(4) = 16/3 - 0 = 16/3.