Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=|x| .\) Graph the functions on the same set of axes. \(g(x)=|x-2|-3\) and \(f(g(x))\)

Step-by-Step Solution

Answer

The function \(f(g(x))\) or ||x – 2| – 3| when graphed will be like a 'W' shape, utilizing the absolute function rules.

Step 1: Understand the given functions

The first function is \(f(x) = |x|\), which is the absolute value of x. The absolute function is a v-shaped graph which intersects the y-axis at (0,0). It has the general equation \(y=a|x-h|+k\), where a, h, and k are constants.The second function is \(g(x) = |x-2|-3\). So for this function, a=1 (since the function is not multiplied by any other number), h=2 (the value within the absolute value brackets that shifts the graph left or right), and k=-3 (a vertical shift of the graph). Thus, this implies that the vertex of the absolute value graph of function g is at (2, -3).

Step 2: Graph g(x) on Set of Axes

Start by graphing \(g(x) = |x-2|-3\). The graph of \(g(x) = |x-2|-3\) is a v-shaped graph with its vertex at the point (2,-3). The graph's slope to the right of that point is +1 and to the left is -1.

Step 3: Construct f(g(x))

Before we graph \(f(g(x))\) we first need to find what this function is. \(f(g(x))\) means that we are substituting \(g(x)\) into \(f(x)\). So if \(f(x) = |x|\), then \(f(g(x)) = |g(x)| = ||x – 2| – 3|\).

Step 4: Graph f(g(x)) on Set of Axes

To graph the function \(f(g(x))\), which is now expressed as \(f(g(x)) = ||x – 2| – 3|\), we can consider the function inside the outer absolute value notation \(|x-2|-3\) as a single entity. This function will not have negative y-values, as its results are the absolute value of g(x). Therefore, for the graph half with y < -3 of \(g(x)\), reflect the graph about the line y = -3 to graph \(f(g(x))\).

Key Concepts

Graphing Absolute Value Functions

Understanding how to graph absolute value functions is fundamental to analyzing their behavior. The absolute value function creates a V-shaped graph, known as its characteristic shape. You start with the basic function, f(x) = |x|, which has a vertex at the origin (0,0). The sloped sides of the graph are mirror images, rising away from the vertex. Now, when we translate this function horizontally or vertically, the vertex moves accordingly but maintains its 'V' shape.

For example, the function g(x) = |x - 2| - 3 will shift the basic 'V' graph to the right by 2 units and downwards by 3 units. The new vertex is therefore at (2, -3). Remember, the graph's slope to the right of the vertex is +1 (upward) and to the left is -1 (also upward due to the absolute value). Keeping this technique in mind will make graphing any absolute value function much simpler.

Composite Functions

Composite functions can be seen as a function within another function, typically denoted as f(g(x)). It’s like putting one function inside the shell of another. To understand this, consider our inner function g(x), and then for each x-value, we find the corresponding g-value, which we then feed into the outer function f.

In our exercise, f(g(x)) means we substitute g(x) into f(x), making our composite function f(g(x)) = ||x – 2| – 3|. It’s really important to work from the inside out. Calculate what happens first in g(x), and then apply the function f to those results. This can significantly alter the graph's appearance, as demonstrated in the example.

Transformation of Functions

Transforming a function refers to moving, stretching, or flipping its graph. The general form of the absolute value function, y = a|x - h| + k, is highly transformable. Here, h and k will shift the graph left/right and up/down, respectively, while a determines the ‘openness’ or ‘steepness’ of the 'V'.

A positive h value moves the graph to the right, while negative moves it left. With k, positive values shift it upwards, negative downwards. If a is greater than 1, the 'V' opens narrower (steeper), and if a is less than 1 but greater than 0, it opens wider. If a is negative, our 'V' flips upside down. Applying these transformations allows us to graph functions like g(x) with ease, simply by 'moving' the basic graph of f(x) around.

Piecewise Functions

Piecewise functions are functions defined by multiple segments or ‘pieces,’ each represented by different rules over different intervals. They are crucial when a single formula cannot describe a situation entirely. For example, an absolute value function can be considered as a piecewise function because it has different equations for the negative and positive parts.

The absolute function g(x) = |x-2| - 3 is essentially two linear functions joined at the vertex. For x < 2, the function is g(x) = -(x - 2) - 3, and for x ≥ 2, it is g(x) = (x - 2) - 3. Analyzing these 'pieces' separately can make graphing more complex functions manageable, as you're dealing with simpler, straight-line segments that you then piece together.