Hello Anthony,
Given f(x) = -2x+1 + 3 , rewrite in x-y notation
y = -2x+1 + 3 , interchange x with y
x = -2y+1 + 3 , isolate the power by subtracting 3
x - 3 = -2y+1 , since the base is 2 and not -2, multiply both sides by -1
-x + 3 = 2y+1, take the common log of both sides
log (-x + 3) = log 2y+1 , according to the log of a power rule we may write the exponent as a factor
log (-x + 3) = (y + 1)*log 2 , distribute the log 2
log (-x + 3) = ylog2 + log2, subtract log 2 on both sides
log (-x +3) - log2 = ylog2, divide both sides by log 2
y = [log (-x+3) - log2] ÷ log2, replacing y with f-1(x)
f-1(x) = [log (-x + 3) - log2] ÷ log2 is the inverse of the the given function f(x).
Since the argument of the log must be positive, it follows that the domain of the inverse function is found by setting the argument of the log greater than zero.
-x + 3 > 0 ⇒ -x > -3 ⇒ x < 3. Hence the domain in interval notation is (-∞, 3).
