Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (f ○ g)(-6)

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation \((f \circ g)(x)\) means \(f(g(x))\). This means you first apply \(g(x)\), and then apply \(f\) to the result.
2. Step 2: Substitute \(-6\) into \(g(x)\). Since \(g(x) = x^2\), calculate \(g(-6) = (-6)^2\).
3. Step 3: Simplify \((-6)^2\) to find the value of \(g(-6)\).
4. Step 4: Substitute the result from Step 3 into \(f(x)\). Since \(f(x) = \sqrt{x-2}\), calculate \(f(g(-6)) = \sqrt{g(-6) - 2}\).
5. Step 5: Simplify \(\sqrt{g(-6) - 2}\) to find \((f \circ g)(-6)\).

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. The notation (f ○ g)(x) means to apply g first and then apply f to the result of g. This requires substituting g(x) into f, allowing us to evaluate the combined function at a specific input.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) = √(x-2), the input must satisfy x-2 ≥ 0, meaning x must be greater than or equal to 2. Understanding the domain is crucial when evaluating composed functions to ensure valid inputs.

Evaluating Functions

Evaluating functions involves substituting a specific value into the function's expression to find the output. In this case, after finding g(-6) = (-6)^2 = 36, we then substitute this result into f. Proper evaluation requires careful arithmetic and adherence to the function's defined operations.