Let $\text{ƒ}\left(x\right)=\text{√}(x-2)$ and $g\left(x\right)=x^2$. Find each of the following, if possible. the domain of $\text{ƒ○}g$

Verified step by step guidance

1. Recall that the composition of functions \( (f \circ g)(x) \) means \( f(g(x)) \). So here, \( (f \circ g)(x) = f(g(x)) = f(x^2) \).
2. Substitute \( g(x) = x^2 \) into \( f(x) = \sqrt{x - 2} \) to get \( f(g(x)) = \sqrt{x^2 - 2} \).
3. To find the domain of \( f \circ g \), determine for which values of \( x \) the expression inside the square root is non-negative, since the square root function requires the radicand to be \( \geq 0 \).
4. Set up the inequality: \( x^2 - 2 \geq 0 \). Solve this inequality to find the values of \( x \) that satisfy it.
5. Solve \( x^2 - 2 \geq 0 \) by isolating \( x^2 \): \( x^2 \geq 2 \). Then find the values of \( x \) such that \( x \leq -\sqrt{2} \) or \( x \geq \sqrt{2} \). These intervals form the domain of \( f \circ g \).

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (ƒ ○ g)(x) = ƒ(g(x)). To find the composition, you substitute g(x) into ƒ(x), which requires understanding how the output of g affects the input of ƒ.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When dealing with compositions, the domain of ƒ ○ g consists of all x-values in the domain of g such that g(x) lies within the domain of ƒ.

Square Root Function Domain Restrictions

For a square root function like ƒ(x) = √(x-2), the expression inside the root must be non-negative. This means x - 2 ≥ 0, so the domain of ƒ is all x ≥ 2. When composing, this restriction applies to g(x), requiring g(x) ≥ 2.