Problem 26 - 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. $$(h \circ f \circ g)(x)$$

Step-by-Step Solution

Answer

\((h \circ f \circ g)(x) = 2\sqrt{x}\) with domain \([0, \infty)\).

Step 1: Understanding Composite Function Notation

The notation \((h \circ f \circ g)(x)\) denotes the composition of three functions: Apply function \(g\) first, then \(f\), and finally \(h\). So we need to find \(h(f(g(x)))\).

Step 2: Evaluate the Inner Function \(g(x)\)

The function \(g(x) = \sqrt{x}\). This function is defined for \(x \geq 0\).

Step 3: Apply the Intermediate Function \(f\)

Next, we apply \(f\) to \(g(x)\):\(f(g(x)) = f(\sqrt{x}) = -2(\sqrt{x}) = -2\sqrt{x}\).This expression is defined for \(x \geq 0\) since it is derived from \(g(x) = \sqrt{x}\).

Step 4: Apply the Outer Function \(h\)

Now, we apply \(h\) to the result of \(f(g(x))\):\(h(f(g(x))) = h(-2\sqrt{x}) = |-2\sqrt{x}|\).The absolute value function \(|x|\) is defined for all real numbers, but given our previous steps, this depends on \(x \geq 0\).

Step 5: Simplify the Expression

We simplify \(|-2\sqrt{x}|\) to \(2\sqrt{x}\), since the absolute value of a non-positive expression \(-2\sqrt{x}\) is simply its positive counterpart. Thus, \((h \circ f \circ g)(x) = 2\sqrt{x}\).

Step 6: Determine the Domain of \(2\sqrt{x}\)

The function \(2\sqrt{x}\) is defined for \(x \geq 0\), as it inherited its domain from \(g(x) = \sqrt{x}\). In interval notation, this domain is \([0, \infty)\).

Key Concepts

Domain of a Function

The domain of a function is the set of input values (usually represented as "x") for which the function is defined. Simply put, it's the range of all possible x-values you can safely plug into a function without getting into trouble, like division by zero or taking the square root of a negative number.
For example, with a function like \(g(x) = \sqrt{x}\), you can't use any negative numbers because the square root of a negative number isn't a real number. Thus, the domain of \(g(x)\) is all real numbers starting from 0 and going to infinity.
To find the domain of a composite function like \(h(f(g(x)))\), you must consider the domains of all involved functions. You begin with the innermost function and ensure that its domain fits within the allowable inputs for the successive functions.
Understanding how the domain is determined helps us write our answer logically in interval notation.

Interval Notation

Interval notation is a clean way of describing a set of numbers or the domain of a function. It's kind of like a shorthand in math.
In interval notation, brackets are used to show which numbers are included in the domain:

• Square brackets \([ ]\) indicate that the endpoints are included.
• Parentheses \(( )\) indicate that the endpoints are not included.

For instance, the interval \([0, \infty)\) shows that all numbers starting from 0 and going to infinity are included, and the 0 is part of the interval. However, \(\infty\) is not a number we can reach, hence we use a parenthesis.
Applying interval notation to a function's domain makes our solutions more precise and universally understandable.

Absolute Value Function

The absolute value function is symbolized with vertical bars, like this: \(|x|\). It measures the distance between a number and zero on a number line, regardless of direction.
Therefore, absolute value always returns a non-negative number.
For example:

• \(|-5| = 5\)
• \(|3| = 3\)

Applying the absolute value function in the expression \( h(f(g(x))) = h(-2\sqrt{x}) \), where applying \( | -2\sqrt{x} | \) leads us to \(2\sqrt{x}\), ensures a non-negative result is always achieved, maintaining the function's definition. This particularly handy feature allows it to fit seamlessly into composite functions.

Square Root Function

The square root function, represented as \(\sqrt{x}\), assigns to each non-negative real number \(x\) a single non-negative number \(y\) such that \(y^2 = x\). It is primarily defined only for non-negative numbers.
In the case of \(g(x) = \sqrt{x}\), the function is defined for all \(x\geq 0\). This keeps things in the realm of real numbers and avoids issues with non-existent numbers in typical real-world math contexts.
When you're using \(\sqrt{x}\) in a composite function (like in \(f(g(x))\)), it's essential to respect this domain, as it restricts which x-values can be used.
Remembering the square root function's characteristics helps correctly sequence calculations in composite functions.