Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the domains of (a) \(f,\) (b) \(g\) and (c) \(f \circ g .\) Use a graphing utility to verify your results. $$f(x)=\sqrt{x+3}, \quad g(x)=\frac{x}{2}$$

Step-by-Step Solution

Answer

The domains of \(f(x)\), \(g(x)\), and \(f \circ g (x)\) are [-3, +\infty), (-\infty,+ \infty) and [-6, +\infty) respectively.

Step 1: Find the Domain of \(f(x)\)

Set the expression inside the square root in \(f(x)\) to be greater or equal to zero, i.e., \(x+3 \geq 0\). Solving this inequality results in \(x \geq -3\). So, the domain of \(f(x)\) is \([-3, +\infty)\].

Step 2: Find the Domain of \(g(x)\)

The function \(g(x) = \frac{x}{2}\) does not have any restrictions. It is defined for all real numbers, so its domain is \(-\infty, +\infty\).

Step 3: Find the Domain of \(f \circ g (x)\)

Substitute \(g(x) = \frac{x}{2}\) into \(f(x) = \sqrt{x+3}\) to get \(f \circ g (x) = \sqrt{\frac{x}{2} + 3}\). The domain of this function is determined by setting the expression inside the square root to be greater than or equal to zero, i.e., \( \frac{x}{2} +3 \geq 0\). Solving this inequality gives \(x \geq -6\). So, the domain of \(f \circ g (x)\) is \([-6, +\infty)\]

Key Concepts

Composite Functions

Composite functions are an essential concept in mathematics where you combine two functions to form a new one. When dealing with composite functions, the notation \( f \circ g \) is used, which means you first apply function \( g \) to a variable and then apply function \( f \) to the result of \( g \).
To determine the value of a composite function at a particular point \( x \), you:

• Substitute \( x \) into the function \( g \).
• Take the result from \( g(x) \) and substitute it into the function \( f \).

In the exercise, the composite function \( f \circ g(x) \) led us to the expression \( \sqrt{\frac{x}{2} + 3} \). It's often necessary to check the domain of composite functions as the domain of \( f \circ g \) is subject to the limitations imposed by both \( f \) and \( g \). The domain must satisfy the conditions required by the square root function, ensuring that the expression inside the root remains non-negative. We concluded that this composite function has a domain of \([-6, +\infty)\).

Inequalities

Inequalities involve expressions that use symbols such as \( >, <, \geq, \) and \( \leq \) to denote relationships between quantities. Understanding inequalities is crucial in solving domain problems, particularly in determining which values a variable can take.
In the example, the function \( f(x) = \sqrt{x+3} \) required setting up an inequality \( x+3 \geq 0 \) to determine the allowed values for \( x \). Solving this inequality gives \( x \geq -3 \), indicating that \( x \) must be greater than or equal to \(-3\), reflecting the domain of the function \( f(x) \).

• To solve inequalities: isolate the variable on one side through valid algebraic operations.
• Keep track of the inequality's direction; it stays the same unless both sides are multiplied or divided by a negative number.

Inequalities are versatile and useful across various mathematical disciplines, often employed to describe ranges of values that satisfy a particular condition.

Square Root Function

The square root function plays a significant role in many mathematical contexts and must meet specific criteria to be defined properly. The function \( f(x) = \sqrt{x} \) is defined only for non-negative values of \( x \) because the square root of a negative number is not real under standard mathematical operations.
To determine the domain for a square root function like \( f(x) = \sqrt{x+3} \):

• Ensure that the expression inside the square root (here, \( x+3 \)) is non-negative by setting up the inequality \( x+3 \geq 0 \).
• Solving this gives \( x \geq -3 \), meaning the domain begins from \(-3\) and extends to positive infinity.

Understanding the nature of the square root function is vital because it affects composite functions and requires careful attention to the signs and values permitted for its input to ensure the function remains defined and real.