Problem 87 - Intermediate Algebra

Question

Identify the domain and then graph each function. \(f(x)=\sqrt{x-3} ;\) use the following table. $$ \begin{array}{|c|c|} \hline x & {f(x)} \\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 7 \\ \hline 12 & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Domain: \([3, \infty)\). Graph starts at (3,0) and curves upwards.

Step 1: Identify the Domain of the Function

The function given is \( f(x) = \sqrt{x-3} \). The square root function is only defined for non-negative values. This means we need \( x-3 \geq 0 \). Solving this inequality for \( x \) gives \( x \geq 3 \). Therefore, the domain of the function is \([3, \infty)\).

Step 2: Calculate Function Values

We will use the domain to determine \( f(x) \) for the values provided in the table:\- For \( x = 3 \), \( f(3) = \sqrt{3-3} = \sqrt{0} = 0 \).\- For \( x = 4 \), \( f(4) = \sqrt{4-3} = \sqrt{1} = 1 \).\- For \( x = 7 \), \( f(7) = \sqrt{7-3} = \sqrt{4} = 2 \).\- For \( x = 12 \), \( f(12) = \sqrt{12-3} = \sqrt{9} = 3 \).\Populate the function values into the table.

Step 3: Populate the Table

Based on the calculations above, fill in the table with the values of the function: $$ \begin{array}{|c|c|} \hline x & {f(x)} \ \hline 3 & 0 \ \hline 4 & 1 \ \hline 7 & 2 \ \hline 12 & 3 \ \hline \end{array} $$

Step 4: Graph the Function

Using the values from the table, plot the points (3,0), (4,1), (7,2), and (12,3) on a coordinate grid. This will form a curve starting at the point (3,0), showing a gradual increase as \( x \) increases. The graph will be a smooth curve that moves upwards and to the right as \( x \) increases within the domain ([3,∞)).

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental concept in mathematics, specifically when working with functions and their graphs. It is denoted as \( f(x) = \sqrt{x} \), where \( \sqrt{x} \) represents the principal square root of \( x \). This means that for any non-negative number \( x \), the function outputs \( y \) such that \( y^2 = x \). For example, if \( x = 4 \), then \( \sqrt{4} = 2 \) since 2 squared equals 4.

One of the key characteristics of the square root function is its domain. The square root is only defined for non-negative inputs (i.e., \( x \geq 0 \)). This is because the square root of a negative number is not a real number. In our specific function \( f(x) = \sqrt{x-3} \), the domain is slightly adjusted to \( x - 3 \geq 0 \), or simply \( x \geq 3 \). Therefore, for this function, the set of permissible inputs starts at 3 and goes to infinity.

Demystifying Inequalities

Inequalities are mathematical expressions involving symbols such as \( >, <, \geq, \leq \), representing relations of size between two values.Understanding inequalities is essential for determining the domain of functions like square roots. In the context of the function \( f(x) = \sqrt{x-3} \), we derive the inequality \( x-3 \geq 0 \). Solving this provides \( x \geq 3 \), meaning any \( x \) value must be greater than or equal to 3 for the function to produce a real number output.

The process of solving inequalities often involves similar steps to solving equations, like adding or subtracting constants and dividing or multiplying both sides by positive numbers. However, it is crucial to reverse the inequality sign when multiplying or dividing by a negative number. This distinguishes inequalities from equations and highlights why they must be handled with care.

Graphing Functions Effectively

Graphing functions is a skill that visually represents mathematical relationships. Understanding how to correctly graph a square root function, like \( f(x) = \sqrt{x-3} \), involves following some straightforward steps:

• Identify points derived from function values, as calculated in the exercise. For instance, our table includes points (3,0), (4,1), (7,2), and (12,3).
• Plot these points on a graph with a coordinate grid. The x-axis represents the input \( x \), and the y-axis shows the output \( f(x) \).
• Connect the points with a smooth curve that increases gradually. This visualizes the function’s behavior as \( x \) increases within its domain, \([3, \infty)\).

Graphing helps in understanding the function’s nature and behavior: a square root function generally has a slow rise and never dips below the x-axis, reflecting its non-negative output characteristic. The resulting graph starts at the domain's boundary point and moves upwards and to the right, confirming the function's growth pattern.