Problem 26 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 34, the function is the set of all ordered pairs \((x, y)\) satisfying the given equation. Find the domain and range of the function, and draw a sketch of the graph of the function. $$ g: y=|x| \cdot|x-1| $$

Step-by-Step Solution

Answer

Domain: (-∞, ∞). Range: [0, ∞).

Step 1 - Understand the Function

Given the function \[ g: y = |x| \cdot |x-1| \] The function is a combination of absolute value expressions \( |x| \) and \( |x-1| \).

Step 2 - Identify the Domain

The domain of a function includes all values of \( x \) for which the function is defined. Here, both \( |x| \) and \( |x-1| \) are defined for all real numbers. Thus, the domain of the function is \[ \text{Domain: } (-\infty, \infty) \]

Step 3 - Determine the Range

Compute the range by examining the possible outputs (\( y \)-values).Consider that both \( |x| \) and \( |x-1| \) are non-negative. Therefore, \( y = |x| \cdot |x-1| \geq 0 \).The minimum value of \( y \) is 0, which occurs at \(x = 0\) or \(x = 1\).Thus, the range of the function is \[ \text{Range: } [0, \infty) \]

Step 4 - Sketch the Graph

To sketch the graph, plot a few key points such as for \( x = 0, y = 0 \), for \( x = 1, y = 0 \), and for \( x = 2, y = 2 \).Draw the curve connecting these points considering that the function is symmetric with respect to |x| and has another symmetry fold from |x-1|.

Key Concepts

Absolute Value Functions

Absolute value functions feature prominently in many mathematical problems. An absolute value function is written as \( |x| \), and represents the distance of a number \( x \) from zero on the number line.
This distance is always non-negative. For example, \( |3| = 3 \) and \( |-3| = 3 \).

In our given function \( g: y = |x| \cdot |x-1| \), we deal with two absolute value expressions: \( |x| \) and \( |x-1| \).
This means for every \( x \), \( |x| \) gives us its non-negative distance from zero, while \( |x-1| \) gives its non-negative distance from 1.
This impacts our graph and the computation of the range because the product of two non-negative values is also non-negative.

Graph Sketching

Graphing the function \( y = |x| \cdot |x-1| \) can be simplified by identifying key points and understanding the symmetry.
First, plot key points such as:

• \( x = 0, y = |0| \cdot |0-1| = 0 \)
• \( x = 1, y = |1| \cdot |1-1| = 0 \)
• \( x = 2, y = |2| \cdot |2-1| = 2 \)

By connecting these points: (0, 0), (1, 0), and (2, 2), we see how the values increase symmetrically on both sides due to the nature of absolute values.

Notice that as \( x \) increases or decreases, \( y \) starts increasing at a rate dictated by the product of the two absolute values.
This means the graph will open upwards, forming a V-shape, but with an additional kink at \( x=1 \), since this is also a zero-point of the function.

Symmetry in Functions

Understanding symmetry in functions can greatly simplify sketching graphs and understanding behavior.
For our function \( g(y) = |x| \cdot |x-1| \), we observe it has symmetry from its absolute value terms.

First, \( |x| \) is symmetric around the y-axis. This means the function will produce the same \( y \)-values for \( x \) and \( -x \). Next, the term \( |x-1| \) suggests another form of symmetry, such as matching values around a different axis or line.

Combining these symmetries means that the overall function \( y \) will be symmetric around both 0 and 1.
This symmetry affects the shape of the graph significantly, suggesting that besides building it step-by-step, one can leverage symmetry principles to sketch it efficiently.
Symmetric properties simplify understanding, ensuring the function behaves similarly on opposite sides of certain points or axes.