Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } x<0 \\ 1-x & \text { if } x>0\end{array}\right.\)

Step-by-Step Solution

Answer

Domain: \((-\infty, 0) \cup (0, \infty)\). Graph includes a parabola and a line.

Step 1: Understand the Function Pieces

The function is piecewise, which means different rules apply to different parts of its domain. Here, we have two pieces: \( f(x) = x^2 \) for \( x < 0 \), and \( f(x) = 1-x \) for \( x > 0 \). There is no function defined at \( x = 0 \).

Step 2: Consider the Domain of Each Piece

For the first piece \( x^2 \), the domain is \( x < 0 \), meaning it is applicable for all negative values of \( x \). For \( 1-x \), the domain is \( x > 0 \), meaning it is applicable for all positive values of \( x \). The function does not include \( x = 0 \).

Step 3: Sketch Each Piece of the Function

For \( x < 0 \), sketch the parabola \( y = x^2 \), which opens upwards and is limited to the left side of the y-axis, approaching but not including \( x = 0 \). For \( x > 0 \), sketch the linear function \( y = 1-x \), which has a negative slope and does not include \( x = 0 \).

Step 4: State the Domain in Interval Notation

The domain is the set of all possible \( x \) values for which the function is defined. Since the function does not include \( x = 0 \), the domain is \( (-\infty, 0) \cup (0, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to the set of input values (usually represented as "x") for which the function is defined. In simpler words, it tells us where the function exists on the x-axis. Since a function might have rules that specify when each part applies, especially in piecewise functions, determining the domain is crucial.
In piecewise functions like the one given in the exercise, different rules apply to different ranges of x values:

• For the piece described by \( x^2 \), the domain includes all x less than 0. This means the function is active in the negative side of the x-axis.
• For the piece \( 1-x \), the domain includes all x greater than 0, which means it exists on the positive side of the x-axis.
• The point \( x = 0 \) is not included since neither function piece has a definition there.

These distinctions collectively give us the full domain in interval notation as \((-\infty, 0) \cup (0, \infty)\). This tells us the function exists at all places except where x equals zero.

Interval Notation

Interval notation is a mathematical shorthand for representing domains and intervals compactly. It uses parentheses and brackets to describe an interval. Knowing how to read and write interval notation is essential for understanding the domain of not just piecewise functions but any kind of function.
In the exercise, the function's domain is expressed as \((-\infty, 0) \cup (0, \infty)\). Let's break this down:

• \((-\infty, 0)\) means all numbers less than 0, but not including 0 itself. Parentheses are used to denote exclusion of the endpoints.
• \((0, \infty)\) means all numbers greater than 0. Again, the parentheses indicate that 0 is not included.
• The "\(\cup\)" symbol means union, which implies the function includes both intervals as part of its domain.

Interval notation helps in quickly visualizing and simplifying the representation of complex and multi-part domains.

Graphing Functions

Graphing functions shows us a visual representation of mathematical relationships. With piecewise functions, it's crucial to understand which portion of the graph applies to each part of the function.
For the exercise, two different types of graphs need to be drawn:

• A parabola for \( x < 0 \)
• A linear function for \( x > 0 \)

When graphing:

• Start by using separate rules for each "piece" of x within their specified intervals.
• For \( x^2 \), draw a parabola opening upwards towards the left of the y-axis. Make sure not to include the value at \( x=0 \).
• For \( 1-x \), draw a straight line starting right after \( x=0 \), decreasing with a negative slope.

Tools like graph paper or graphing calculators can help with accuracy, but understanding the behavior of each piece is most important.

Parabola

A parabola is a symmetric curve shaped like an arch. It's a key feature of quadratic functions, which are represented by equations like \( y = x^2 \).
In the context of the piecewise function from the exercise, we are interested in the parabolic aspect when \( x < 0 \):

• The equation \( y = x^2 \) describes a parabola that opens upwards.
• For \( x < 0 \), this parabola exists only on the left side of the y-axis.
• As it approaches \( x = 0 \), it gets closer to the axis but never touches the value where \( x = 0 \) is defined because \( x \) can never be zero in this part.
• Parabolas are characterized by being smooth curves, and understanding their direction and steepness can help in graphing.

Understanding parabolas helps with several disciplines, not just mathematics, because of their appearance in physics and engineering.

Linear Function

Linear functions are the most basic mathematical models represented graphically as straight lines. They show a constant rate of change and are typically represented as \( y = mx + b \).
In the piecewise function from the exercise, the linear component is described by \( 1-x \), applicable only for \( x > 0 \):

• The equation \( y = 1-x \) is a linear function with a slope of -1, meaning it decreases uniformly as x increases.
• This straight line exists solely on the positive side of the x-axis, beginning right above zero.
• Unlike curves, linear functions do not bend. They go straight from one point to another, making them easier to predict over their interval.

These characteristics make linear functions great for modeling straightforward scenarios in mathematics and real-world applications.