Problem 27 - Precalculus

Question

Use \(f(x)=-2 x, g(x)=\sqrt{x}\) and \(h(x)=|x|\) to find and simplify expressions for the following functions and state the domain of each using interval notation. $$(g \circ f \circ h)(x)$$

Step-by-Step Solution

Answer

The expression is \(g(f(h(x))) = \sqrt{-2|x|}\) with domain \([0, 0]\).

Step 1: Understand the Composition of Functions

The function \((g \circ f \circ h)(x)\) represents the composition of three functions: first apply \(h(x)\), then \(f(x)\), and finally \(g(x)\). Thus, it can be written as \(g(f(h(x)))\).

Step 2: Apply the Function h(x)

The function \(h(x)\) is the absolute value function, which outputs the absolute value of \(x\). So, \(h(x) = |x|\). Therefore, the result of applying \(h\) is \(|x|\).

Step 3: Apply the Function f(x) to the Result of h(x)

Next, apply \(f(x)\), which is \(-2x\), to the output of \(h(x)\). Substituting \(|x|\) into \(f(x)\), we have: \(f(h(x)) = f(|x|) = -2|x|\).

Step 4: Apply the Function g(x) to the Result of f(h(x))

Finally, apply \(g(x)\), which is \(\sqrt{x}\), to the output of \(f(h(x))\). Substituting \(-2|x|\) into \(g(x)\), we have: \(g(f(h(x))) = g(-2|x|) = \sqrt{-2|x|}\).

Step 5: Determine the Domain

The domain of \(g(f(h(x))) = \sqrt{-2|x|}\) requires the expression inside the square root to be non-negative for real number outputs. However, \(-2|x|\) is negative or zero, meaning this simplifies the expression to only work when it's exactly zero, that is, \(x = 0\). Thus, the domain is \([0, 0]\).

Key Concepts

Domain of a Function

The domain of a function refers to all possible input values (usually denoted as "x") for which the function is well-defined and produces valid outputs. For a function composition like

• \((g \circ f \circ h)(x)\),

we must consider the domain of each function involved:- **First**, determine the set of inputs the initial function can accept.- **Second**, ensure outputs from the first function are valid inputs for the next function in the composition.- This chain continues until all functions in the composition are considered.
To find the domain, it is essential to identify constraints like undefined values or domain restrictions that arise from operations within each function, such as division by zero, negative numbers under a square root, etc. As we analyze the given functions, our function composition must adhere to these rules to be valid.

Absolute Value Function

The function \(h(x) = |x|\) is known as the absolute value function. It converts any input \(x\) to its non-negative equivalent. This means that regardless of whether \(x\) is positive, negative, or zero, the output will always be non-negative. In practical terms, the absolute value function serves several unique purposes in calculations:

• It simplifies expressions that involve distances. For example, the absolute value of a number can be seen as its distance from zero on the number line.
• This characteristic is helpful when the direction of a number is not important, only its magnitude.

When using absolute values in function compositions, as in

• \(h(x)\) as in
\( (g \circ f \circ h)(x) \), it implies that any negative inputs will be converted to positive before any further operations occur.

This can affect the outcome of following functions such as \(f(x) = -2x\). This is because these outputs are based on the non-negative versions of the inputs.

Square Root Function

The function \(g(x) = \sqrt{x}\) is known as the square root function. It calculates the square root of a non-negative number, returning only the principal (non-negative) square root. This function has some critical properties and domain restrictions:

• The square root function is only defined for values of \(x\) that are zero or positive.
• Negative values under the square root sign do not result in real numbers.

In the context of our function composition

• \((g \circ f \circ h)(x)\),

placing a negative number like

• \(-2|x|\)

under the square root normally would not be valid. However, since absolute values are non-negative, we usually can proceed. But if any operation makes the expression inside the square root negative or zero overall, that indicates a potential restriction on the domain.

Interval Notation

Interval notation is a mathematical method to denote a set of numbers along the number line. It uses parentheses and brackets to show the nature of boundaries:

• Parentheses \((\) or \()\) mean the endpoint is not included (open interval).
• Brackets \([\) or \()]\) mean the endpoint is included (closed interval).

For example,

• [a, b] includes all numbers from \(a\) to \(b\),
• while (a, b) includes all numbers between \(a\) and \(b\), not including \(a\) and \(b\).

In the case of

• \(\sqrt{-2|x|}\), the domain was determined to be
• \([0, 0]\)

, implying that the function is only defined or returns a valid result when \(x = 0\). Using correct interval notation helps accurately represent the constraints and limitations of function domains.