Problem 65 - College Algebra with Modeling and Visualization

Question

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x+4, \quad g(x)=\sqrt{4-x^{2}} $$

Step-by-Step Solution

Answer

(a) \(f(g(x))=\sqrt{4-x^2}+4\), domain: \([-2, 2]\), (b) \(g(f(x))=\sqrt{4-(x+4)^2}\), domain: \([-6, -2]\), (c) \(f(f(x))=x+8\), domain: \(\mathbb{R}\).

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

The composition \((f \circ g)(x)\) means we need to substitute \(g(x)\) into \(f(x)\). Given \(f(x) = x + 4\) and \(g(x) = \sqrt{4-x^2}\), substitute \(g(x)\) into \(f(x)\):\[(f \circ g)(x) = f(g(x)) = g(x) + 4 = \sqrt{4 - x^2} + 4.\]For the domain, \(g(x) = \sqrt{4-x^2}\) is only defined when \(4 - x^2 \geq 0\), or \(-2 \leq x \leq 2\). Thus, the domain of \((f \circ g)(x)\) is \([-2, 2]\).

Step 2: Finding \((g \circ f)(x)\)

The composition \((g \circ f)(x)\) means we need to substitute \(f(x)\) into \(g(x)\). Thus, substitute \(f(x) = x + 4\) into \(g(x)\):\[(g \circ f)(x) = g(f(x)) = \sqrt{4 - (x+4)^2}.\]To find the domain, solve \(4 - (x+4)^2 \geq 0\). Expanding, \(x + 4\) gives \[(x+4)^2 = x^2 + 8x + 16.\]So we need \(4 \geq x^2 + 8x + 16\). Rearrange this to form \(x^2 + 8x + 12 \leq 0\), factor to get \((x+2)(x+6) \leq 0\). Solve this inequality: \(-6 \leq x \leq -2\). Hence, the domain of \((g \circ f)(x)\) is \([-6, -2]\).

Step 3: Finding \((f \circ f)(x)\)

The composition \((f \circ f)(x)\) means substitute \(f(x)\) into itself. Substitute \(f(x) = x + 4\) again into \(f(x)\):\[(f \circ f)(x) = f(f(x)) = (x+4) + 4 = x + 8.\]This is a linear function with domain \( \mathbb{R} \) (all real numbers), since there are no restrictions from the operation.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when working with mathematical expressions. The domain refers to all the possible input values (usually denoted as 'x') for which the function is defined.
In simpler terms, it's the set of all x-values that you can plug into the function without breaking any mathematical rules.

For example, consider the function \( g(x) = \sqrt{4-x^2} \). The expression inside the square root, \( 4-x^2 \), must be greater than or equal to zero. This is because the square root of a negative number is not defined in the set of real numbers.

• To find the domain, we solve the inequality \( 4 - x^2 \geq 0 \).
• This inequality can be solved to find that \( -2 \leq x \leq 2 \).

This means that the domain of the function is from -2 to 2, including both endpoints.

Linear Functions

Linear functions are among the simplest types of functions. They form a straight line when plotted on a graph. A general form of a linear function is given by \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In our exercise, \( f(x) = x + 4 \) is a linear function with:

• Slope \( m = 1 \), meaning it rises one unit for every unit it travels across.
• Y-intercept \( b = 4 \), indicating that it crosses the y-axis at 4.

One crucial aspect of linear functions is that they are defined for all real numbers. This means their domain is the entire set of real numbers, denoted by \( \mathbb{R} \). In the context of this exercise, this property is exemplified by \( f(x) = x + 4 \) and its self-composition \( (f \circ f)(x) = x + 8 \). For both cases, the domain remains \( \mathbb{R} \).

Square Root Function

The square root function doesn’t accept just any input because of the properties of square roots. A function like \( g(x) = \sqrt{4-x^2} \) emphasizes this point.

The expression \( 4 - x^2 \) under the root must not be negative, adhering to the rule that square roots of negative numbers are not real.

• As already discussed, solving \( 4 - x^2 \geq 0 \) yields \( -2 \leq x \leq 2 \).

Thus, a square root function is only defined for inputs that keep the value inside the root non-negative. This ensures the results are real, not imaginary.

Inequalities in Algebra

Inequalities help to find the range of variable values that satisfy a given condition. They're frequently used to determine the domain of functions, especially when involving roots or rational expressions.

The solution of an inequality, like \( 4 - x^2 \geq 0 \), involves:

• Solving \( x^2 - 4 \leq 0 \) by finding the x-values that make the expression true.
• Factoring, if possible, like transforming \( x^2 + 8x + 12 \leq 0 \) to \( (x+2)(x+6) \leq 0 \).

These steps help find intervals within which inequality holds true. Intervals provide valuable insights into the behavior and constraints of functions, assisting in identifying permissible domains and relationships between elements in algebra.