find the inverse.
f(x)=-2+log_4(x-4)
To find the inverse of the function f(x) = -2 + log₄(x - 4), we'll follow these steps:
Step 1: Replace f(x) with y.
y = -2 + log₄(x - 4)
Step 2: Swap x and y.
x = -2 + log₄(y - 4)
Step 3: Solve for y.
Rearrange the equation to solve for y:
x + 2 = log₄(y - 4)
Step 4: Rewrite in exponential form.
Rewrite the equation in exponential form:
4^(x + 2) = y - 4
Step 5: Solve for y.
Add 4 to both sides of the equation:
4^(x + 2) + 4 = y
Step 6: Simplify the equation.
Combine like terms on the left side:
y = 4^(x + 2) + 4
Therefore, the inverse of the function f(x) = -2 + log₄(x - 4) is given by g(x) = 4^(x + 2) + 4.