Problem 30 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find \(f(x)\) for the indicated values of \(x\), if possible. (b) Find the domain of \(f\). $$ f(x)=\sqrt{x-3} \text { for } x=4, a+4 $$

Step-by-Step Solution

Answer

\(f(4) = 1\), \(f(a+4) = \sqrt{a+1}\); Domain: \(x \geq 3\).

Step 1: Calculate f(x) for x = 4

First, substitute 4 into the function \(f(x) = \sqrt{x - 3}\). \[ f(4) = \sqrt{4 - 3} = \sqrt{1} = 1 \]

Step 2: Calculate f(x) for x = a + 4

Substitute \(a + 4\) into the function \(f(x) = \sqrt{x - 3}\). \[ f(a+4) = \sqrt{(a+4) - 3} = \sqrt{a + 1} \] The expression is only defined when \(a + 1 \geq 0\) to have a real number output, meaning \(a \geq -1\).

Step 3: Determine the Domain of f(x)

The function \(f(x) = \sqrt{x - 3}\) is only defined when the expression inside the square root is non-negative. Set \(x - 3 \geq 0\) and solve for \(x\): \[ x - 3 \geq 0 \] Adding 3 to both sides gives: \[ x \geq 3 \] Therefore, the domain of \(f\) is all real numbers \(x\) such that \(x \geq 3\).

Key Concepts

Function Evaluation

Function evaluation involves substituting specific values into a given function to determine the corresponding output. This process is crucial in understanding how functions relate inputs to outputs through a defined mathematical expression.
Imagine having a function, which is like a machine. You put something in and get a result out. For example, if you have the function \(f(x) = \sqrt{x - 3}\), to evaluate \(f(x)\) for a specific number, you substitute that number in place of \(x\).

• The function \(f(x)\) is defined as \(f(x) = \sqrt{x - 3}\).
• To calculate \(f(4)\), replace \(x\) with 4 in the expression: \(f(4) = \sqrt{4 - 3}\).
• Simplify the equation to find \(f(4) = \sqrt{1} = 1\).

Function evaluation can also involve algebraic expressions, such as \(x = a + 4\). In this case, you replace \(x\) with \(a + 4\), resulting in \(f(a+4) = \sqrt{a+1}\). This expression is defined only if \(a + 1\) is non-negative, which leads to the condition \(a \geq -1\).
Understanding how to substitute values into functions and simplify is a fundamental skill in algebra and calculus.

Square Root Function

The square root function is a specific type of function that involves finding the square root of a value or expression. It is represented as \(f(x) = \sqrt{x}\). The square root function provides real number results only for non-negative arguments.
Understanding how square root functions behave is important, especially when they are part of more complex expressions like \(f(x) = \sqrt{x - 3}\). The expression inside the square root must be zero or positive to ensure a real output. When evaluating a square root function, always consider:

• The square root of a positive number is a real number.
• The square root of zero is zero.
• The square root of a negative number is not a real number.

For \(f(x) = \sqrt{x - 3}\), you first need to ensure the expression \(x - 3\) is non-negative, meaning \(x \geq 3\). Without a non-negative value under the square root, the function does not produce a real number solution. This helps define the function's domain, restricting it to inputs that allow sensible real number outputs.

Real Number Domain

The domain of a function refers to the set of all possible input values (\(x\)) that the function can accept without causing mathematical contradictions, like taking the square root of a negative number. For real number domains, the goal is to determine when the function produces valid outputs across all real numbers.
When dealing with a square root function like \(f(x) = \sqrt{x - 3}\), the domain is set based on ensuring the expression inside the square root is zero or positive. Solve \(x - 3 \geq 0\) for \(x\) to determine the domain:

• Add 3 to both sides: \(x \geq 3\).

This tells you that \(f(x)\) can only accept values of \(x\) starting from 3 and going to infinity. Consequently, the domain is presented as all real numbers \(x\) such that \(x \geq 3\). By understanding this, you ensure that the function does not try to evaluate a square root of a negative number, which would result in undefined outputs in the real number system.