Problem 34 - A Graphical Approach to College Algebra

Question

Use the results of the specified exercises to determine (a) the domain and (b) the range of each function. \(y=|x-4|-3\) (Exercise 22)

Step-by-Step Solution

Answer

Domain: All real numbers; Range: \( y \ge -3 \).

Step 1: Analyze the Basic Function

The given function is \( y = |x-4| - 3 \). The basic absolute value function is \( y = |x| \), which is defined for all real numbers. This means its domain is all real numbers, \( x \in \mathbb{R} \).

Step 2: Determine the Domain of the Given Function

The function \( y = |x-4| - 3 \) is simply a transformation of \( y = |x| \). The inside of the absolute value, \( x-4 \), shifts the graph horizontally but does not affect its domain. Therefore, the domain of \( y = |x-4| - 3 \) remains all real numbers, \( x \in \mathbb{R} \).

Step 3: Analyze Vertical Shift for Range

Next, analyze the range by considering the transformation. The function \( |x-4| \) reaches a minimum value of 0 when \( x = 4 \). When 0, the function becomes \( y = -3 \).

Step 4: Determine the Range of the Function

Given that \( y = |x-4| - 3 \) can be expressed as \( y = |x-4| - 3 = 0 - 3 = -3 \) at its minimum and it increases without bound as \( x \) moves away from 4, the smallest value is \(-3\). Hence, the range is \( y \ge -3 \).

Key Concepts

Domain of a Function

The domain of a function is essentially the set of all possible input values that the function can accept. For most functions, this is defined as all real numbers that can be plugged into the function without causing any mathematical issues like division by zero or taking the square root of a negative number. In the case of an absolute value function such as \( y = |x-4| - 3 \), the domain remains all real numbers.

This occurs because absolute value functions do not have restrictions like division or square roots that can limit input values. In simpler terms, you can insert any real number for \( x \) and still get a valid output. So, for the given function, the domain is all real numbers, represented mathematically as \( x \in \mathbb{R} \). Understanding the function's domain is crucial since it tells you which inputs are valid for equations and graphs.

Range of a Function

The range of a function concerns the possible output values that a function can produce. If you imagine a graph of a function, the range refers to the vertical spread of the graph in the coordinate plane. For an absolute value function like \( y = |x-4| - 3 \), the range is influenced by the transformations applied to the basic function \( y = |x| \).

A vertical shift is present here, which moves the whole graph down by 3 units. This means the smallest potential value for \( y \) occurs when \( x = 4 \), creating a minimum of \( y = -3 \). Because the function can increase indefinitely as \( x \) moves away from 4, the outputs cover all values of \( y \) greater than or equal to -3. Therefore, the range of this function is expressed as \( y \ge -3 \). Understanding the range is vital for predicting what the function can output, which is useful in solving real-world problems.

Absolute Value Function

An absolute value function is a specific type of function defined as \( f(x) = |x| \), where \(|x|\) denotes the absolute value of \( x \). This mathematical operation provides the distance of \( x \) from zero, regardless of whether \( x \) is positive or negative, thus always yielding non-negative results.

The graph of a basic absolute value function is V-shaped, opening upwards, and it is symmetric about the y-axis. Such functions are impactful in various areas, including real-life situations that require handling negative values consistently, such as calculating deviations in statistics. In transformations, an absolute value function can shift up, down, or sideways. For example, the function \( y = |x-4| - 3 \) is horizontally shifted 4 units to the right from the origin and 3 units downward.

Understanding the characteristics of absolute value functions can help in better analyzing their behavior and applications in different scenarios.