In Exercises 101–102, find an equation for f^(-1)(x). Then graph f and f^(-1) in the same rectangular coordinate system. f(x) = 1 - x^2, x ≥ 0.

Verified step by step guidance

1. Identify the function given, which is f(x) = 1 - x^2, and note the domain restriction x ≥ 0.
2. To find the inverse function, f^(-1)(x), start by replacing f(x) with y. So, y = 1 - x^2.
3. Swap x and y to begin solving for the new y, which will give us x = 1 - y^2.
4. Solve for y by isolating y on one side of the equation. This involves taking the square root of both sides, remembering to consider only the positive root since the original function's domain is x ≥ 0. Thus, y = sqrt(1 - x).
5. Replace y with f^(-1)(x) to denote the inverse function. Therefore, f^(-1)(x) = sqrt(1 - x), with the domain of f^(-1)(x) being 0 ≤ x ≤ 1, corresponding to the range of the original function.

Key Concepts

Inverse Functions

An inverse function, denoted as f^(-1)(x), reverses the effect of the original function f(x). For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input. To find the inverse, we typically swap the x and y variables in the equation and solve for y.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For the function f(x) = 1 - x^2 with the restriction x ≥ 0, the domain is [0, ∞) and the range is (-∞, 1]. Understanding the domain and range is crucial when finding the inverse, as they will switch roles.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the relationship between the input and output values. When graphing f(x) and its inverse f^(-1)(x), it is important to note that the graphs are reflections over the line y = x. This visual representation helps in understanding how the original function and its inverse relate to each other.