Let $\text{ƒ}\left(x\right)=\text{√}(x-2)$ and $g\left(x\right)=x^2$. Find each of the following, if possible
$\left(g\text{○ƒ}\right)\left(3\right)$

Verified step by step guidance

1. Understand that the notation \((g \circ f)(3)\) means we first apply the function \(f\) to 3, then apply the function \(g\) to the result of \(f(3)\).
2. Evaluate \(f(3)\) by substituting \(x = 3\) into the function \(f(x) = \sqrt{x - 2}\). This gives \(f(3) = \sqrt{3 - 2}\).
3. Simplify the expression inside the square root to find the value of \(f(3)\).
4. Next, use the value found for \(f(3)\) as the input for the function \(g(x) = x^2\). So, calculate \(g(f(3)) = (f(3))^2\).
5. Simplify the expression \((f(3))^2\) to find the final value of \((g \circ f)(3)\).

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (g ○ f)(x) = g(f(x)). It requires evaluating the inner function first, then using that output as the input for the outer function. Understanding this process is essential to correctly compute composite functions.

Domain of a Function

The domain is the set of all input values for which a function is defined. For functions involving square roots, the expression inside must be non-negative to avoid imaginary numbers. Checking the domain ensures that the function values are real and the composition is valid.

Evaluating Functions at Specific Points

Evaluating a function at a specific point means substituting the input value into the function's formula and simplifying. This step is crucial when finding the value of composite functions at given inputs, as it involves careful substitution and arithmetic.