Problem 88 - College Algebra with Corequisite Support

Question

For the following exercises, find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). \((g \circ f)(a) ; \quad(f \circ g)(a)\)

Step-by-Step Solution

Answer

The composition \\(g \circ f(a)\\) does not exist for all a, and \\(f \circ g(a)\\) exists if \\(a \geq 2\\).

Step 1: Understand the notation

The notation \(g \circ f\)(a) means to take the function \(f(x)\), plug it into the function \(g(x)\), and evaluate at \(a\). Similarly, \(f \circ g(a)\) means to take \(g(x)\), plug it into \(f(x)\), and evaluate at \(a\).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The composition \\(g \circ f(a)\\) does not exist for all a, and \\(f \circ g(a)\\) exists if \\(a \geq 2\\).

Key Concepts

Domain of Functions

The domain of a function represents all the possible input values (or "x" values) that a function can accept without making any illegal operations, like dividing by zero or taking the square root of a negative number. For the given functions:
- **Function \(f(x) = x^2 + 2\)**: Here, \(f(x)\) is a polynomial function. The domain of \(f(x)\) is specified as all \(x \geq 0\). This means that only non-negative numbers can be input into this function.
- **Function \(g(x) = \sqrt{x - 2}\)**: This function involves a square root. The expression under the square root, \(x - 2\), must be non-negative (\(x - 2 \geq 0\)). Thus, the domain of \(g(x)\) is all \(x \geq 2\).
When composing functions, you need to ensure the output of the first function falls within the domain of the second function. This consideration is crucial when determining valid inputs for the composite functions \((g \circ f)(x)\) and \((f \circ g)(x)\).

Function Notation

Function notation helps us understand and perform operations with functions more easily. It simplifies the representation of complex operations, such as function composition.
- **\((g \circ f)(a)\)** reads as "g composed with f at a." This means you first compute \(f(a)\), then use the result as the input for \(g(x)\).- **\((f \circ g)(a)\)** means "f composed with g at a." Here, you start by calculating \(g(a)\), and the resulting value becomes the input for \(f(x)\).
Using function notation correctly is vital in composition because it directs the sequence of applying the functions. Always pay attention to which function is listed first since it significantly affects the final result. Remember, composing functions is not commutative, meaning \((g \circ f)(x)\) is generally not equal to \((f \circ g)(x)\).

Step-by-Step Solution

When solving a function composition problem step by step, it is important to follow a systematic approach:
**Step 1**: Identify and understand the notation used. We have two compositions here: \((g \circ f)(a)\) and \((f \circ g)(a)\).
**Step 2**: Begin with \((g \circ f)(a)\). Calculate \(f(a)\) using the formula for \(f(x)\), and then substitute this result into \(g(x)\). Ensure that the result of \(f(a)\) lies within the domain of \(g(x)\).
**Step 3**: To solve \((f \circ g)(a)\), first compute \(g(a)\) by substituting \(a\) into the formula for \(g(x)\), and then use this result as the input for \(f(x)\). Again, verify that the resultant value of \(g(a)\) is acceptable under the domain of \(f(x)\).
Following these steps ensures clarity and accuracy in your solutions. Always check the domains to confirm your input values are valid for the operations involved. Step-by-step problem-solving not only refines your understanding but also uncovers potential pitfalls in function compositions.