Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0

Verified step by step guidance

1. Start with the given function: \(y = f(x) = x^{2} + 1\) where \(x \geq 0\).
2. To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x^{2} + 1\).
3. Swap the variables \(x\) and \(y\) to find the inverse: \(x = y^{2} + 1\).
4. Solve this equation for \(y\): subtract 1 from both sides to get \(x - 1 = y^{2}\).
5. Since \(x \geq 0\) in the original function, take the positive square root to get \(y = \sqrt{x - 1}\), which is the formula for \(f^{-1}(x)\).

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. If y = f(x), then x = f⁻¹(y). The graph of an inverse function is a reflection of the original function's graph across the line y = x.

Domain and Range Restrictions

To find an inverse function, the original function must be one-to-one, often requiring domain restrictions. Here, f(x) = x² + 1 is restricted to x ≥ 0 to ensure it is invertible, as the parabola is not one-to-one over all real numbers.

Finding the Inverse Function Algebraically

To find f⁻¹(x), replace f(x) with y, swap x and y, then solve for y. For f(x) = x² + 1 with x ≥ 0, swapping gives x = y² + 1, so y = √(x - 1), which matches the inverse function shown in the graph.