Problem 41 - Algebra for College Students

Question

\(f(x)=\sqrt{3 x-4}\)

Step-by-Step Solution

Answer

The domain is \( x \geq \frac{4}{3} \).

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt{3x-4} \). This is a square root function where the expression \( 3x - 4 \) is under the square root sign. The domain of this function is determined by the values of \( x \) that make the expression inside the square root non-negative because the square root of a negative number is undefined.

Step 2: Determine the Domain

To find the domain, solve the inequality \( 3x - 4 \geq 0 \). This ensures that the expression inside the square root is non-negative.

Step 3: Solve the Inequality

Rearrange the inequality to find the values of \( x \). Add 4 to both sides to obtain \( 3x \geq 4 \). Then, divide both sides by 3 to isolate \( x \), resulting in \( x \geq \frac{4}{3} \).

Step 4: Conclusion on Domain

The domain of \( f(x) = \sqrt{3x-4} \) is all \( x \) such that \( x \geq \frac{4}{3} \). This means \( x \) must be greater than or equal to \( \frac{4}{3} \) for \( f(x) \) to be defined.

Key Concepts

Square Root Function

The square root function is a type of function involving a square root symbol, which is often denoted as \( \sqrt{\cdot} \). When the function includes the square root operation, it means we are looking at the square root of whatever expression is inside the symbol. For instance, in the function \( f(x) = \sqrt{3x-4} \), the expression \( 3x-4 \) is inside the square root symbol.

Square root functions have particular features:

• The square root itself only yields non-negative values, which means the expression inside the square root must also be non-negative for the function to be mathematically valid and defined.
• They usually shape the graph into a curve that's one-half of a parabola, starting from a certain point and extending horizontally.
• The domain of the square root function is limited to values that keep the expression inside the root zero or more, ensuring that the square root is proper and defined.

Understanding these aspects helps predict the behavior of the square root function graphically and algebraically.

Inequality Solving

Inequality solving is a mathematical process used to find the set of values that satisfy an inequality condition. An inequality shows that one expression is either less than or greater than another, but not necessarily equal. To solve an inequality means to determine which values make this condition true.

For the inequality \( 3x-4 \geq 0 \), it means finding values of \( x \) that keep the expression \( 3x-4 \) non-negative. Solving it involves:

• Rearranging the inequality: Start by moving constant terms to one side, like adding 4 to both sides of \( 3x-4 \geq 0 \) to get \( 3x \geq 4 \).
• Isolating the variable: Divide by the coefficient of \( x \), here divide by 3, resulting in \( x \geq \frac{4}{3} \).

When solving inequalities, remember:

• You can add or subtract the same number from both sides without flipping the inequality sign.
• When multiplying or dividing by a negative number, the inequality sign flips direction.

This ensures that the solutions to the inequality accurately reflect all possible values that satisfy the original inequality.

Function Definition

Every function has a specific definition that explains how input values (the domain) are transformed into output values (the range) through a rule or formula. In the function \( f(x) = \sqrt{3x-4} \), the rule is the operation that combines the variable \( x \) with the expression \( 3x-4 \) under a square root.

When defining a function, consider these key elements:

• Domain Definition: The domain includes all possible input values for the function. Here, because the expression inside the square root must be non-negative, \( x \) values must satisfy \( x \geq \frac{4}{3} \).
• Output or Range: The resulting values after applying the function rule to the domain. Since the square root of a number gives a non-negative result, the range of this function is all non-negative numbers.
• Rule/Application: The formula used to compute outputs from inputs, like determining \( f(x) \) by substituting \( x \) in \( \sqrt{3x-4} \).

In essence, a function is a mathematical relationship expressing a dependency between elements from two sets—the set of inputs and their corresponding outputs.