Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Use graphing to determine the domain and range of \(y=f(x)\) and of \(y=|f(x)|\). $$f(x)=-|x+2|-2$$

Step-by-Step Solution

Answer

Domain of both functions: \((-\infty, \infty)\). Range of \( f(x) \): \((-\infty, -2]\), Range of \(|f(x)|\): \([0, \infty)\).

Step 1: Understand the Function

First, identify the function given: \( f(x) = -|x+2| - 2 \). This is an absolute value function that has been reflected vertically and shifted downwards. Absolute value functions have a V-shape. In this case, the negative sign in front of \(|x+2|\) indicates that the V-shape will open downward.

Step 2: Determine the Domain of \( f(x) \)

The domain of an absolute value function, including transformations like shifts or reflections, is all real numbers. Therefore, the domain of \( f(x) = -|x+2|-2 \) is \( (-\infty, \infty) \).

Step 3: Determine the Range of \( f(x) \)

The vertex of the downward-opening V-shape is the maximum point of the function. For \( f(x) = -|x+2|-2 \), the vertex occurs at \( x = -2 \). At this point, \( f(-2) = -0 - 2 = -2 \). Thus, the range of \( f(x) \) is \(( -\infty, -2] \).

Step 4: Graph \( f(x) \)

Sketch the graph of \( f(x) = -|x+2|-2 \) using its basic properties: an opening downwards V-shape with a vertex at \(( -2, -2) \). This visual will aid in analyzing the transformation to \( y = |f(x)| \).

Step 5: Analyze \( y = |f(x)| \)

For \( y = |f(x)| \), any negative output of \( f(x) \) becomes positive. This results in reflecting all parts of the graph of \( f(x) \) that lie below the x-axis, onto their corresponding parts above the x-axis.

Step 6: Determine the Domain of \( y = |f(x)| \)

The domain remains unchanged when taking the absolute value of a function. Hence, the domain of \( y = |f(x)| \) is also \( (-\infty, \infty) \).

Step 7: Determine the Range of \( y = |f(x)| \)

The smallest value of \( |f(x)| \) is 0, occurring at \( x = -2 \) when \( f(-2) = -2 \). Thus, the range of \( y = |f(x)| \) starts at 0 and includes all positive numbers: \( [0, \infty) \).

Step 8: Finalize the Domain and Range

Combine the results: for \( f(x) = -|x+2|-2 \), the domain is \( (-\infty, \infty) \) and the range is \(( -\infty, -2] \). For \( y = |f(x)| \), the domain is also \( (-\infty, \infty) \), but the range is \( [0, \infty) \).

Key Concepts

Domain and Range

In mathematical terms, the domain of a function includes all the possible input values (x-values) that will result in a well-defined function output. For the function given, \( f(x) = -|x+2|-2 \), any real number can be plugged into the equation and will provide a result, which implies that the domain is all real numbers. We can express this in interval notation as \((-\infty, \infty)\).

• The domain of \( y = |f(x)| \) also remains \((-\infty, \infty)\) because the absolute value function, being a modification of our original function, does not restrict any x-values.

The range refers to all the potential output values (y-values) after plugging in the domain into the function. For the original given function \( f(x) \), it is clear that it can never output anything larger than -2 because of the negative reflection and vertical shift. Therefore, the range is \((-\infty, -2]\).

• Concerning \( y=|f(x)| \), since we are taking the absolute value of the outputs of the original function, the negative values below the x-axis flip to positive. Thus, the lowest value is now 0, resulting in a range of \([0, \infty)\).

Absolute Value Functions

Absolute value functions are mathematical functions expressed in the format \( y = |x| \). Their characteristic V-shape graph is due to their nature of returning non-negative output for any input value. In simpler terms, the absolute value of a number reflects its distance from zero on a number line, without considering its sign.

• For \( f(x) = -|x+2|-2 \), the absolute value expression \(|x+2|\) is shifted and reflected.
• The inclusion of a negative sign (-) before the absolute value flips the V-shape graph downwards, essentially a vertical reflection.

When graphing \( y = |f(x)| \), the transformation involves flipping any negative parts of the graph up, as the absolute value function essentially cannot yield negative results after taking the absolute value of f(x).

Function Transformations

Function transformations alter the basic graph of a function in various ways including shifts, reflections, stretches, and compressions. Let’s look at \( f(x) = -|x+2|-2 \):

• Horizontal Shift: The term \(x+2\) suggests that the entire graph shifts to the left by 2 units.
• Vertical Shift: The constant -2 results in a downward shift by 2 units on the y-axis.
• Reflection: The negative sign (-) reflects the graph across the x-axis, causing its typical upward V-shape to flip downwards.

When encountering such transformations, it’s helpful to take note of any additional modifications to understand how the graph will look and how it will influence domain and range.

• For \( y = |f(x)| \), the outcome of the function being an absolute value means, f(x) reflecting about the x-axis always results in non-negative y-values.

Precalculus Concepts

Precalculus serves as the foundation for higher-level math, involving exploration of functions, algebra, and trigonometry. Understanding precalculus is crucial as it preps you for studying calculus.

• Key skills include analyzing and graphing functions such as the absolute value function, recognizing transformations, and comprehending domains and ranges.
• By comprehending these transformations, one develops an insight into how functions behave in the wider world of mathematics and beyond.
• Graphing is a particularly vital skill, allowing you to visualize and predict the behavior of complex functions effectively.

This knowledge is not only fundamental to solving mathematical problems but also forms the basis for applications in sciences and engineering fields.