Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = x²+3, x≥0; g(x) = √x-3, x≥3

Verified step by step guidance

1. Recall that two functions \( f \) and \( g \) are inverses if and only if \( f(g(x)) = x \) for all \( x \) in the domain of \( g \), and \( g(f(x)) = x \) for all \( x \) in the domain of \( f \).
2. Start by finding \( f(g(x)) \). Substitute \( g(x) = \sqrt{x} - 3 \) into \( f(x) = x^2 + 3 \), so \( f(g(x)) = (\sqrt{x} - 3)^2 + 3 \).
3. Next, simplify \( f(g(x)) \) by expanding the square: \( (\sqrt{x} - 3)^2 = (\sqrt{x})^2 - 2 \cdot 3 \cdot \sqrt{x} + 3^2 = x - 6\sqrt{x} + 9 \). Then add 3 to get \( f(g(x)) = x - 6\sqrt{x} + 12 \).
4. Now, find \( g(f(x)) \). Substitute \( f(x) = x^2 + 3 \) into \( g(x) = \sqrt{x} - 3 \), so \( g(f(x)) = \sqrt{x^2 + 3} - 3 \).
5. Check if \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true for all \( x \) in their respective domains. If both equalities hold, then \( f \) and \( g \) are inverses; otherwise, they are not.

Key Concepts

Definition of Inverse Functions

Two functions f and g are inverses if applying one after the other returns the original input, meaning f(g(x)) = x and g(f(x)) = x for all x in their domains. This relationship shows that each function 'undoes' the effect of the other.

Domain and Range Restrictions

When determining inverses, it is crucial to consider the domain and range of each function. Restrictions like x ≥ 0 for f and x ≥ 3 for g ensure the functions are one-to-one and their inverses are well-defined, preventing ambiguity in outputs.

Composition of Functions

Function composition involves substituting one function into another, such as f(g(x)) or g(f(x)). Checking if these compositions simplify to the identity function (x) confirms whether two functions are inverses, making composition a key tool in this verification.