Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-34: Find (a) \((f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\). $$ f(x)=\sqrt{3-x}, \quad g(x)=\sqrt{x^{2}-16} $$

Step-by-Step Solution

Answer

(a) \((f \circ g)(x) = \sqrt{3-\sqrt{x^2-16}}\) with domain \([-5, -4] \cup [4, 5]\). (b) \((g \circ f)(x) = \sqrt{-x - 13}\) with no valid domain.

Step 1: Find \((f \circ g)(x)\)

To find \((f \circ g)(x)\), we must substitute \(g(x)\) into \(f(x)\). We have \(f(x) = \sqrt{3-x}\) and \(g(x) = \sqrt{x^2-16}\). Therefore,\[ (f \circ g)(x) = f(g(x)) = \sqrt{3-\sqrt{x^2-16}} \]

Step 2: Determine the domain of \(g(x)\)

The function \(g(x) = \sqrt{x^2-16}\) is defined when the expression under the square root is non-negative. Thus, solve:\[ x^2 - 16 \geq 0 \]This implies:\[ (x-4)(x+4) \geq 0 \]The solution is \(x \leq -4\) or \(x \geq 4\). So, the domain of \(g(x)\) is \((\infty, -4] \cup [4, \infty)\).

Step 3: Determine the domain of \(f(x)\) for \(f(g(x))\)

The function \(f(x) = \sqrt{3-x}\) is defined when the expression under the square root is non-negative, so:\[ 3 - \sqrt{x^2-16} \geq 0 \]This implies:\[ \sqrt{x^2-16} \leq 3 \]Squaring both sides, we have:\[ x^2 - 16 \leq 9 \]So, \(x^2 \leq 25\), which means:\[-5 \leq x \leq 5\].

Step 4: Combine domains to find domain of \(f \circ g\)

Combine the domains of \(g(x)\) and \(f(x)\) to find the domain of \(f \circ g\). We need \(g(x)\) to be defined and \(f(g(x))\) to be defined:- \(x \leq -4\) or \(x \geq 4\) from \(g(x)\)- \(-5 \leq x \leq 5\) from \(f(g(x))\)The intersection of these intervals is \([-5, -4] \cup [4, 5]\). Therefore, the domain of \(f \circ g\) is \([-5, -4] \cup [4, 5]\).

Step 5: Find \((g \circ f)(x)\)

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). We have \(g(x) = \sqrt{x^2-16}\) and \(f(x) = \sqrt{3-x}\). Thus,\[ (g \circ f)(x) = g(f(x)) = \sqrt{(\sqrt{3-x})^2 - 16}\]Simplifying by squaring the square root:\[ (g \circ f)(x) = \sqrt{3-x-16} = \sqrt{-x-13} \]

Step 6: Determine the domain of \(f(x)\)

\(f(x) = \sqrt{3-x}\) is defined when \(3-x \geq 0\). Solving gives:\[ x \leq 3 \]Thus, the domain of \(f(x)\) is \((-\infty, 3]\).

Step 7: Determine the domain of \(g(f(x))\)

\(g(f(x)) = \sqrt{-x-13}\) is defined when \(-x-13 \geq 0\). Solving gives:\[ -x \geq 13 \Rightarrow x \leq -13\]However, \(x\) must also be in the domain of \(f(x)\), \(-\infty, 3]\), so there is no intersection. Thus, the domain of \(g \circ f\) is empty.

Key Concepts

Function Composition

Function composition is an operation that takes two functions and combines them into a single function. This process involves taking the output of one function and using it as the input for another function. Imagine you have functions \( f(x) \) and \( g(x) \). If you compose \( f \) with \( g \), you get a new function denoted as \( (f \circ g)(x) = f(g(x)) \). Similarly, composing \( g \) with \( f \) yields \( (g \circ f)(x) = g(f(x)) \).
For a simple analogy, think of it like a factory process where the output (product) of the first machine (function) becomes the raw material (input) for the second machine. Function composition is often used in mathematics to simplify problems or to create new functions that capture a sequence of operations efficiently. Just like combination ingredients to cook a meal, function composition allows us to delve deeper into how functions interact and create complex relationships. It's essential to pay attention to each step carefully, ensuring each function's range can serve as the domain of the other to have a well-defined composition.

Domain of a Function

The domain of a function is the set of all input values, often denoted as \( x \), for which the function is defined. Understanding a function's domain is crucial because it tells us the conditions under which a function operates correctly.
When determining a function's domain, consider:

• Values that would lead to division by zero.
• Negative values under an even root, such as a square root, which would result in non-real numbers.
• Any other restrictions given by the nature of the function.

For instance, the domain of the square root function \( \sqrt{x} \) is \( x \geq 0 \) because square roots of negative numbers are not real.
In composition, the domain is often influenced by multiple functions. For \( (f \circ g)(x) \), the domain consists of inputs \( x \) where \( g(x) \) is defined, and \( f(g(x)) \) is also defined. This requires checking the domain of \( g(x) \) first and ensuring its output fits the domain criteria of \( f(x) \). This dual verification ensures that the composed function is valid.

Square Root Function

The square root function, represented as \( \sqrt{x} \), is a fundamental mathematical function that returns the principal square root of \( x \). This means it provides the non-negative number that, when multiplied by itself, equals \( x \).
Key properties include:

• It is only defined for non-negative numbers, so its domain is \( x \geq 0 \).
• The graph of the square root function is a smooth curve beginning at the origin \((0,0)\) and extending to infinity as \( x \) increases.
• It is a non-linear function, meaning it does not form a straight line.

When dealing with compositions involving square root functions, it's crucial to ensure all expressions under the square root are non-negative. This can lead to inequalities that specify a valid range for the input \( x \).
For example, in the function \( f(x)=\sqrt{3-x} \), the requirement \( 3-x \geq 0 \) implies \( x \leq 3 \). Careful attention to these details helps in identifying the appropriate domain for function compositions.

Inequalities in Functions

Inequalities play a significant role in determining the domain of functions, especially those involving square roots or other complex operations. When a function like \( \sqrt{x} \) is part of an equation, ensuring the expression inside the square root remains non-negative leads to essential inequality solutions.
Consider the function \( g(x) = \sqrt{x^2-16} \). To determine when it's defined, we must solve the inequality \( x^2-16 \geq 0 \). This simplifies to \( (x-4)(x+4) \geq 0 \), indicating that \( x \leq -4 \) or \( x \geq 4 \).
Steps to solve such inequalities often involve:

• Setting the expression inside the root \( \geq 0 \).
• Factoring the quadratic expression, when possible.
• Identifying the intervals that satisfy the inequality, known as the solution set.

These steps defined the valid input range for the function, ensuring the output is real and valid. In composite functions, combining these domain restrictions is necessary to determine the overall domain effectively, as seen in the solution process of \( (f \circ g)(x) \) and \( (g \circ f)(x) \). An empty domain indicates no value of \( x \) will satisfy the combined conditions, which can occur if the compositions impose mutually exclusive restrictions.