Problem 63 - Intermediate Algebra

Question

Find each function value, if possible. Do not use a calculator. See Example 5. $$ f(x)=\sqrt{3 x+1} $$ a. \(f(8)\) b. \(f(-2)\)

Step-by-Step Solution

Answer

\( f(8) = 5 \); \( f(-2) \) is undefined.

Step 1: Understand the function

The given function is \( f(x) = \sqrt{3x + 1} \). It involves a square root, which requires the expression inside to be non-negative because the square root of a negative number is undefined in the real number system.

Step 2: Evaluate \( f(8) \)

Substitute \( x = 8 \) into the function: \( f(8) = \sqrt{3(8) + 1} \). Calculate the expression inside the square root: \( 3(8) + 1 = 24 + 1 = 25 \). So, \( f(8) = \sqrt{25} \). Since the square root of 25 is 5, \( f(8) = 5 \).

Step 3: Check the domain for \( f(-2) \)

Before substituting \( x = -2 \) into the function, check if \( 3x + 1 \geq 0 \). Substitute \( x = -2 \): \( 3(-2) + 1 = -6 + 1 = -5 \), which is negative. Since the expression inside the square root is negative, \( f(-2) \) is undefined.

Key Concepts

Understanding the Square Root Function

When we talk about a square root function, we are dealing with expressions that have a square root symbol, \( \sqrt{} \), involved. In the given function \( f(x) = \sqrt{3x + 1} \), the square root implies that we are looking for a number that, when squared, gives us the expression \( 3x + 1 \).
The square root function has some special characteristics:

• It often results in two possible values (positive and negative) in the domain of real numbers, but in most mathematics contexts, we only consider the principal (positive) square root.
• The expression inside the square root, called the radicand, must be non-negative to ensure the function yields real number results.
• The symbol \( \sqrt{} \) is denoted specifically for non-negative outputs of the radicand.

This function requires us to first solve the equation under the square root to ensure it makes sense within real numbers.

Defining the Domain of a Function

A function's domain is essentially all the possible input values (commonly denoted as \( x \)) for which the function can be evaluated. When dealing with functions that involve square roots, understanding the domain becomes crucial because these functions are only defined when their radicand is non-negative.
For the function \( f(x) = \sqrt{3x + 1} \), the domain is determined by solving the inequality \( 3x + 1 \geq 0 \). This gives us:

• Subtract 1 from both sides to get \( 3x \geq -1 \).
• Divide each side by 3, resulting in \( x \geq -\frac{1}{3} \).

Thus, the domain of \( f(x) \) includes all real numbers \( x \) such that \( x \geq -\frac{1}{3} \). Understanding this helps prevent errors such as trying to calculate the function value where it is not defined.

Avoiding Undefined Expressions

Undefined expressions occur when calculations or functions attempt to deal with numbers or operations outside of their defined scope. In our case, the trouble arises when the radicand in a square root function is negative.
In the realm of real numbers:

• Expressions inside a square root must be zero or positive \( (\geq 0) \) since the square root of a negative number is not defined.
• This understanding prevents us from making incorrect calculations or assumptions about a function's outputs.

For instance, when evaluating \( f(-2) \) for \( f(x) = \sqrt{3x + 1} \), substituting \( x = -2 \) results in the radicand \( 3(-2) + 1 = -5 \), which is negative. Hence, \( f(-2) \) is undefined, reflecting the importance of checking the function's domain before plugging in values.