Given f(x)=-4+lnx; Find the inverse of f. f^-1=?
original:
y = 4 + lnx
inverse:
x = 4 + lny
solving for y
lny = x-4
y = e^(x-4)
f^-1 (x) = e^(x-4)
How do you find the domain and range?
domain is your choice of x's that you can use in your equation
For logs, one of the main properties is that I can only take the log of a positive number
so for y = 4 + lnx , the domain is x > 0
however, the result of taking such logs results in any real number, so the range is y ∈ R
After taking the inverse of a function, the domain of the original becomes the range of the inverse, and the range of the original becomes the domain of the inverse.
To find the inverse of the function f(x) = -4 + ln(x), we need to follow these steps:
Step 1: Replace f(x) with y, which gives us the equation: y = -4 + ln(x).
Step 2: Swap the variables x and y: x = -4 + ln(y).
Step 3: Solve the equation for y. Let's do that:
First, we isolate the natural logarithm term by adding 4 to both sides: x + 4 = ln(y).
Next, we eliminate the natural logarithm by taking the exponential of both sides. Since the exponential function is the inverse of the natural logarithm, we have: e^(x + 4) = y.
Therefore, the inverse of f(x) = -4 + ln(x) is f^(-1)(x) = e^(x + 4).