Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. $$y=|x-4|-3$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\). Range: \([-3, \infty)\).

Step 1: Understanding the Function

The function given is \( y = |x-4| - 3 \). It combines the absolute value function \( |x-4| \) and a downward vertical shift by 3 units. Absolute value functions create a V-shape and are defined for all real numbers.

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the absolute value function \( |x-4| \), there are no restrictions on the values \( x \) can take, as any real number can be used. Thus, the domain is all real numbers, expressed as \( (-\infty, \infty) \).

Step 3: Determine the Range

The range is the set of all possible output values (y-values) of a function. The basic absolute value function \( |x| \) has a minimum value of 0. For \( |x-4| \), this minimum value occurs when \( x=4 \). Subtracting 3 shifts the minimum value to \( -3 \). Therefore, the range of \( y = |x-4| - 3 \) is \( [-3, \infty) \), as the function outputs values starting from \(-3\) and increasing without bound.

Key Concepts

Absolute Value Function

The absolute value function, written as \(|x|\), is a mathematical expression that describes the distance of a number \(x\) from zero, regardless of its direction on the number line. This function creates a V-shaped graph that is symmetric about the y-axis, and its vertex, or the lowest point, is at (0,0). The basic property of absolute values is that they are always non-negative.

• For \(|x - 4|\), the expression shifts the vertex of the V-shape from (0,0) to (4,0).
• This tells us that the lowest point of this new function is when \(x = 4\).

Recognizing this shift is crucial in understanding how the absolute value function influences each term in a mathematical expression, especially when further manipulated by vertical shifts or other operations.
An absolute value function allows for all real numbers as inputs, which means it does not have any restrictions on its domain. This makes it versatile in various mathematical scenarios.

Real Numbers

Real numbers encompass all the numbers on the number line, including all the rational numbers, such as 0, -7, 3.5, and \(\frac{2}{3}\), as well as the irrational numbers, such as \(\sqrt{2}\) and \(\pi\). They are essentially any number that can be found in the real world and represented as a value.

• In functions like \(y = |x - 4| - 3\), each \(x\) in the domain can be any real number.
• This wide range allows the absolute value function to be applied freely without any constraints on the input values.

When analyzing the domain of a function related to real numbers, it's important to note that the domain of absolute value functions, like many basic functions, is indeed all real numbers, often expressed in interval notation as \((-\infty, \infty)\). This means we can input any real number into the function, and it will output a real number.

Vertical Shift

A vertical shift involves moving a graph up or down along the y-axis. In mathematical terms, if you have a function \(f(x)\), applying a vertical shift gives you a new function \(f(x) + C\) or \(f(x) - C\), where \(C\) is a real number.
For the function \(y = |x - 4| - 3\), the \(-3\) indicates a vertical shift downwards by 3 units.

• This shift affects the range by reducing the minimum value of the function by 3 units.
• The lowest point of the original absolute value function \(|x - 4|\) was at 0, so the vertical shift transforms it to -3.

This means for the given function, while the domain remains the same, covering all real numbers, the range is affected. The range, therefore, becomes \([-3, \infty)\), because the lowest value the function can reach is now \(-3\) after applying the vertical shift. Understanding vertical shifts is crucial because they change how and where the function appears on the coordinate plane without altering the function's overall shape.