Use the definition of inverses to determine whether ƒ and g are inverses. f(x) = x^2+3, x≥0; g(x) = √x-3, x≥3

Verified step by step guidance

1. Step 1: To determine if two functions \( f \) and \( g \) are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).
2. Step 2: Start by finding \( f(g(x)) \). Substitute \( g(x) = \sqrt{x-3} \) into \( f(x) = x^2 + 3 \). This gives \( f(g(x)) = (\sqrt{x-3})^2 + 3 \).
3. Step 3: Simplify \( f(g(x)) \). The expression \( (\sqrt{x-3})^2 \) simplifies to \( x-3 \), so \( f(g(x)) = x-3 + 3 = x \).
4. Step 4: Now, find \( g(f(x)) \). Substitute \( f(x) = x^2 + 3 \) into \( g(x) = \sqrt{x-3} \). This gives \( g(f(x)) = \sqrt{(x^2 + 3) - 3} \).
5. Step 5: Simplify \( g(f(x)) \). The expression \( (x^2 + 3) - 3 \) simplifies to \( x^2 \), so \( g(f(x)) = \sqrt{x^2} = x \), given \( x \geq 0 \). Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), \( f \) and \( g \) are inverses.

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. For functions f and g to be inverses, applying g to the output of f should return the original input, and vice versa. Mathematically, this means f(g(x)) = x and g(f(x)) = x for all x in the domain of the respective functions.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). When determining if two functions are inverses, it is crucial to consider their domains and ranges, as they must align appropriately for the inverse relationship to hold.

Composition of Functions

The composition of functions involves combining two functions to create a new function. For functions f and g, the composition f(g(x)) means substituting g(x) into f. To verify if f and g are inverses, we need to compute both f(g(x)) and g(f(x)) and check if both equal x, confirming their inverse relationship.