To find the inverse of the function f(x) = x² – 25, we will follow a systematic approach.
Step 1: Replace f(x) with y.
We start by rewriting the function as:
y = x² – 25
Step 2: Swap x and y.
To find the inverse function, we swap the roles of x and y:
x = y² – 25
Step 3: Solve for y.
Next, we solve this equation for y:
- Add 25 to both sides:
- x + 25 = y²
- Next, take the square root of both sides to isolate y:
- y = ±√(x + 25)
The solution contains both the positive and negative square roots because the original function is a quadratic, which means it does not have a single-valued inverse unless we restrict its domain. Typically, one would restrict the domain of the original function to either negative or positive values of x to ensure that the inverse is a function in the traditional sense.
Step 4: State the inverse function.
If we restrict the domain of the original function to x ≥ 0, the inverse function can be expressed as:
f-1(x) = √(x + 25)
If we restrict to x < 0, we would have:
f-1(x) = -√(x + 25)
In conclusion, the inverse of f(x) = x² – 25 is:
- f-1(x) = √(x + 25) (if x ≥ 0)
- f-1(x) = -√(x + 25) (if x < 0)
This will depend on the domain you choose to work with.