Problem 64 - A Graphical Approach to College Algebra

Question

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=x+4, \quad g(x)=\sqrt{4-x^{2}}$$

Step-by-Step Solution

Answer

(a) Domain: \([-2, 2]\); (b) Domain: None; (c) Domain: \((- fty, fty)\).

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

First, substitute \( g(x) = \sqrt{4-x^2} \) into \( f(x) \). This gives:\[(f \circ g)(x) = f(g(x)) = f(\sqrt{4-x^2}) = \sqrt{4-x^2} + 4\]

Step 2: Identify the domain of \( (f \circ g)(x) \)

The expression \( \sqrt{4-x^2} \) is defined where \(4-x^2 \geq 0\). Solving this inequality:\[4-x^2 \geq 0 \implies 4 \geq x^2 \implies -2 \leq x \leq 2\]Thus, the domain of \( (f \circ g)(x) \) is \([-2, 2]\).

Step 3: Determine \( (g \circ f)(x) \)

Next, substitute \( f(x) = x+4 \) into \( g(x) \). This gives:\[(g \circ f)(x) = g(f(x)) = g(x+4) = \sqrt{4 - (x+4)^2}\]Simplify to:\[\sqrt{4 - (x^2 + 8x + 16)} = \sqrt{-x^2 - 8x - 12}\]

Step 4: Identify the domain of \( (g \circ f)(x) \)

The expression \( \sqrt{-x^2 - 8x - 12} \) is defined where \(-x^2 - 8x - 12 \geq 0\). Solving for:\[-x^2 - 8x - 12 \geq 0 \]The solution requires finding the roots of the quadratic using the quadratic formula. Solving for \(x\), the roots are complex since the discriminant \((-8)^2 - 4(-1)(-12) < 0\), which means \( g(f(x)) \) has no real number domain. Thus, the domain is empty.

Step 5: Determine \( (f \circ f)(x) \)

Substitute \( f(x) = x+4 \) into itself:\[(f \circ f)(x) = f(f(x)) = f(x+4) = (x+4) + 4 = x + 8\]

Step 6: Identify the domain of \( (f \circ f)(x) \)

The function \( x + 8 \) is a linear function, which is defined for all real numbers. Therefore, the domain of \( (f \circ f)(x) \) is all real numbers \((-fty, fty)\).

Key Concepts

Domain of a Function

When talking about functions, the domain refers to all the possible input values for which the function is defined and produces a real output. Understanding the domain is crucial, as it defines the boundaries within which you can use the function.
The domain can sometimes be all real numbers, like with linear functions, but at other times, it might be restricted. For instance, consider any function with a square root. The expression inside the root must be non-negative since the square root of negative numbers leads to imaginary results. This is why when dealing with something like \( \sqrt{4-x^2} \), the valid domain comes from solving \( 4-x^2 \geq 0 \), ensuring you get real numbers. In our example, this requirement resulted in the domain \([-2, 2]\).

• Always solve inequalities to find domains with square roots.
• For simple expressions like \( x+8 \), the domain is all real numbers because there are no restrictions.

By determining the domains correctly, you ensure functions work as intended and avoid mathematical errors.

Composite Functions

Composite functions involve combining two functions, applying one function to the result of another. In simpler terms, if you have two functions, say \( f(x) \) and \( g(x) \), you can create a composite function like \( (f \circ g)(x) \), meaning you apply \( g \) first and then apply \( f \) to the result.
To build a composite function, you substitute one function's expression in place of the variable in another function. For example, to find \( (f \circ g)(x) \), take \( g(x) \) and plug it into \( f(x) \). This can sometimes alter domains, based on how the functions interact when combined. In the example provided, \( (f \circ g)(x) = \sqrt{4-x^2} + 4 \) required checking \( \sqrt{4-x^2} \), leading us to limit \( x \) to be between \(-2\) and \(2\).

• Start with the function inside the parentheses when creating composite functions.
• Check domains after forming a composite function to capture any new restrictions.

Understanding composite functions is fundamental to grasp how different processes can be linked in mathematics, often leading to more complex outcomes.

Quadratic Inequalities

Quadratic inequalities are expressions where a quadratic form, such as \( ax^2 + bx + c \), is set in relation to \(0\) using inequality symbols like \(\geq\) or \(\leq\). Solving these inequalities helps in finding the domain of certain functions.
Here, the process involves determining where the quadratic expression is greater than or equal to zero (or less, depending on the problem). For example, solving \( 4-x^2 \geq 0 \) leads to the domain \([-2, 2]\) for \( (f \circ g)(x) \).

• Factor or use the quadratic formula to find roots of the related equation \( ax^2 + bx + c = 0 \).
• Test intervals determined by the roots to see where the inequality holds.

Using these intervals effectively delineates where the physical or applicable range exists for the function or inequality, a crucial aspect of analysis in these problems.