Given f(x)=3(4^x), what is f^−1 (x)?
y = 3 * 4^x
becomes
x = 3 * 4^y
4^y = x/3
y log 4 = log (x/3) = log x - log 3
y = (log x - log 3) / log 4
f(x)=3(4^x)
or
y = 3(4^x)
step1: to form the inverse equation, interchange the x and y
x = 3(4^y)
x/3 = 4^y
take log of both sides
log(x/3) = log(4^y)
logx - log3 = ylog4
y = (logx - log3)/log4 ---> f^-1 (x) = (logx - log3)/log4
To find the inverse of a function, f^−1(x), we need to swap the roles of x and y and solve for y.
Step 1: Start with the given function, f(x) = 3(4^x).
Let y = f(x), so we have y = 3(4^x).
Step 2: Swap x and y to obtain the inverse equation:
x = 3(4^y).
Step 3: Solve for y:
Divide both sides by 3 to isolate the exponent:
x/3 = 4^y.
Step 4: Take the logarithm (base 4) of both sides:
log base 4 (x/3) = y.
Step 5: Replace y with f^−1(x):
f^−1(x) = log base 4 (x/3).
Therefore, the inverse of the function f(x) = 3(4^x) is f^−1(x) = log base 4 (x/3).