Problem 42 - College Algebra

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

The domain is \((-\infty, 0) \cup (0, \infty)\).

Step 1: Identify the piecewise components

The function is divided into two pieces: \( f(x) = x^2 \) for \( x < 0 \), and \( f(x) = 1 - x \) for \( x > 0 \). Each piece corresponds to a specific range of \( x \) values.

Step 2: Define the domain of each component

The domain for \( f(x) = x^2 \) is \( x < 0 \). The domain for \( f(x) = 1 - x \) is \( x > 0 \). The entire domain of the piecewise function is the combination of both these intervals.

Step 3: Graph each piece of the function separately

First, graph \( f(x) = x^2 \) only for \( x < 0 \). This is a parabola opening upwards, displayed on the left side of the graph since it applies to negative \( x \)-values. Then, graph \( f(x) = 1 - x \) only for \( x > 0 \). This is a straight line with a negative slope, starting on the positive side of the x-axis.

Step 4: Identify and mark key points

The function does not include \( x = 0 \), so there is no value or point to plot exactly at \( x = 0 \). Indicate this by leaving an open circle at \( x = 0 \) for each piece, showing that these points do not belong to the function.

Step 5: Determine the complete domain

Combine the domains of the separate parts into interval notation for the whole function: \( (-\infty, 0) \cup (0, \infty) \). This reflects that all real numbers except \( x = 0 \) are included in the domain.

Key Concepts

Domain in Interval Notation

When determining the domain of a function, we are pinpointing all valid inputs (or x-values) for which the function can produce an output.
In interval notation, we express this set of x-values using parentheses and brackets, quickly conveying whether a value is included or not.

For our exercise, the function is piecewise and comprises two parts, each having its own domain.

• First part: The parabola, represented by \(f(x) = x^2\), applies for \(x < 0\).
  This means we only consider negative values of x, which correspond to the interval \((-\infty, 0)\).
• Second part: The linear function, \(f(x) = 1 - x\), is valid for \(x > 0\), indicating that x-values greater than zero are in the interval \((0, \infty)\).

By merging both domains, we cover all x-values except at zero, leading to the overall domain of the piecewise function: \((-\infty, 0) \cup (0, \infty)\).
Remember, open circles in the graph at \(x = 0\) emphasize it is not part of the domain.

Graphing Functions

Graphing functions means plotting different parts of a function on the coordinate system to visually understand their behavior.
Each part of a piecewise function is plotted according to its conditions, helping visualize how different rules apply to different x-ranges.

For a piecewise function like in our example, begin by:

• Drawing the graph for \(f(x) = x^2\) when \(x < 0\).
  This is a typical parabola starting at negative infinity and approaching zero, curving upwards towards the x-axis as it nears zero.
• Next, focus on \(f(x) = 1 - x\) for \(x > 0\).
  This part is a line decreasing steadily, starting from \(x = 0\) and moving infinitely right, diving below the x-axis.
  Use its slope to determine the angle of descent.

Notably, at \(x = 0\), neither piece is defined, so use open circles to signify this missing point in each graph section.

Parabolas

A parabola is a U-shaped curve that represents quadratic functions like \(f(x) = x^2\).
In any parabola, the vertex is the point where it changes direction, creating its characteristic shape.

The parabola in our example opens upwards.

• It's defined only for negative x-values, which means it doesn’t have the full U shape here—just the left half.
  The curve starts from negative infinity and moves upward toward zero, remaining entirely on the negative x-side.
• The values of the function continuously decrease and approach zero as the x-value increases from further negative to closer to zero.

Because x = 0 is not included in the function's domain, this parabola never actually reaches or touches zero, which is noted by the open circle at the point closest to zero on the x-axis.

Linear Functions

Linear functions are straightforward—they form straight lines on a graph and can be expressed in the form \(f(x) = mx + b\), where m represents the slope.
This slope is crucial as it tells us how steep the line is.

In our piecewise function, the linear part is \(f(x) = 1 - x\), where the slope \(m\) equals -1.

• Because the slope is negative, the line descends as it moves from left to right.
  The y-intercept is at \(y = 1\), meaning this line crosses the y-axis at this point.
• As only x > 0 is included, begin graphing from an open circle at \(x = 0\).
  Extend the line to the right, crossing below the x-axis as it moves further out.

With a slope of -1, the line is equally steep regardless of the point.
Every move right by 1 unit results in the graph moving 1 unit downward.