Understanding the Inverse Function
To find the inverse of the function f(x) = x² + 16, we need to follow several steps that involve swapping the roles of the input and output and solving for the new input.
Step-by-Step Process
- Start with the function: y = x² + 16
- Switch x and y: x = y² + 16
- Now solve for y:
- Subtract 16 from both sides: x – 16 = y²
- Take the square root of both sides:
- y = ±√(x – 16)
Identifying the Inverse Function
The result we have, y = ±√(x – 16), reflects two branches: the positive and negative square roots. For the inverse to be entirely functioning as a proper function, we need to restrict the domain of the original function.
The original function f(x) = x² + 16 is a parabola that opens upwards and has its vertex at (0, 16). To simplify the inverse function and ensure it passes the vertical line test, we can restrict x to non-negative values (i.e., x ≥ 0).
Final Inverse Function
Therefore, with the restriction in place, the inverse function can be expressed as:
f-1(x) = √(x – 16)
This version of the inverse function is valid for x ≥ 16 to ensure that we are working with real numbers.