Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Find the domain and range of the function. $$g(x)=|2 x+3|$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = |2x+3|\) is \(x \in (-\infty, +\infty)\) and the range is \(y \in [0, +\infty)\).

Step 1: Graph the function

Using a graphing utility, plot the function \(g(x)=|2x+3|\). The graph of this function is a V-shape with the vertex at the point where the function equals zero.

Step 2: Find the domain

The domain of a function is the set of all possible x-values which will make the function 'work', and will output real y-values. Looking at our graph, we see that the graph extends to positive and negative infinity along the x-axis. Therefore, the domain of this function is \(x \in (-\infty, +\infty)\). Thus the function is defined for all real numbers.

Step 3: Find the range

The range of a function is the set of possible output values (y-values), which result from using the function formula. From the graph, we can observe that the lowest point of the 'V' is at y=0 and the graph extends upwards to positive infinity. Thus, the range of this function is \(y \in [0, +\infty)\). The function always outputs values greater than or equal to zero

Key Concepts

Graphing Utility

A graphing utility is a handy tool, often in the form of an online application, graphing calculator, or software, that helps visualize mathematical functions. It can be particularly useful when dealing with more complex functions like absolute value functions. Instead of plotting each point by hand, a graphing utility can create precise graphs quickly, allowing you to see the overall shape and key features of the function.

When you use a graphing utility to plot the function \( g(x) = |2x + 3| \), you'll notice that the graph has a characteristic V-shape. This V-shape is common in absolute value functions. The vertex of this V can be found at the point where the expression inside the absolute value equals zero. This point is crucial in understanding the function's behavior, as it often represents the minimum or maximum value on the graph depending on the function's orientation.

Domain and Range

Understanding the domain and range is essential to fully grasp the behavior of a function.

• Domain: The domain of a function represents all the possible input values (x-values) that the function can accept. For the function \( g(x) = |2x + 3| \), the domain is all real numbers. This is because no matter what value you plug in for \( x \), you will get a defined output. The graph shows this by extending infinitely in both directions along the x-axis.


• Range: The range, on the other hand, is all the possible output values (y-values) the function can produce. In our specific function \( g(x) = |2x + 3| \), the lowest y-value occurs at the vertex, \( y = 0 \). Since the absolute value always yields non-negative results, the graph extends upwards towards infinity. Therefore, the range of this function is \( y \in [0, +\infty) \).

Real Numbers

Real numbers consist of all the numbers that can be found on the number line. This includes both rational numbers (like fractions and terminating or repeating decimals) and irrational numbers (like \( \sqrt{2} \) or Pi).

In the context of our function, \( g(x) = |2x + 3| \), both its domain and range are defined in terms of real numbers. The domain includes all real numbers because any real number can be substituted for \( x \), ensuring the function produces a real output.

The range starts from zero and extends through all positive real numbers, reflecting the nature of the absolute value which never yields a negative result. Understanding real numbers helps in identifying how functions behave, and ensures clarity when discussing values that a function can take or produce.