Problem 27 - Algebra for College Students

Question

\(f(x)=\left\\{\begin{aligned} x & \text { for } x \geq 0 \\ 3 x & \text { for } x<0 \end{aligned}\right.\)

Step-by-Step Solution

Answer

At \( x \geq 0 \), \( f(x) = x \); at \( x < 0 \), \( f(x) = 3x \).

Step 1: Identify the intervals for the piecewise function

The function \( f(x) \) is a piecewise function. This means it has different expressions for different intervals of \( x \). Before analyzing the function in detail, let's first identify these intervals: 1. When \( x \geq 0 \), the function \( f(x) = x \).2. When \( x < 0 \), the function \( f(x) = 3x \).

Step 2: Analyze the function for \( x \geq 0 \)

For values of \( x \geq 0 \), the function is defined as \( f(x) = x \). This simply means that the output of the function is equal to the input value. If you input any number that is 0 or positive, the output will be the same number.

Step 3: Analyze the function for \( x < 0 \)

For values of \( x < 0 \), the function is defined as \( f(x) = 3x \). This means that for any negative value of \( x \), the output of the function is three times the input value. For example, if you input \( x = -2 \), then \( f(-2) = 3(-2) = -6 \).

Step 4: Understand the function values at transition points

A transition between the pieces of this function occurs at \( x = 0 \). At this point, we apply \( f(x) = x \) because the condition for \( x \geq 0 \) is fulfilled. Therefore, \( f(0) = 0 \). This ensures the function is continuous at this transition point from non-negative to negative \( x \).

Key Concepts

Intervals of Functions

Understanding intervals is essential when dealing with piecewise functions. Each interval represents a portion of the domain where a specific expression or rule applies. For the given piecewise function \(f(x)=\begin{cases} x & \text{for } x \geq 0 \ 3x & \text{for } x<0 \end{cases}\), we identify two different intervals.

• When \(x \geq 0\), the function is governed by the rule \(f(x) = x\).
• When \(x < 0\), the function follows the expression \(f(x) = 3x\).

The intervals divide the domain of the function based on conditions applied to the input variable, \(x\). Recognizing these separate intervals is the first step in analyzing how the function behaves over its entire domain.

Function Continuity

Continuity of a function refers to whether a function has any abrupt changes or gaps. For piecewise functions, it is crucial to check at transition points, where the expressions change.
In this function, a potential transition point is at \(x = 0\). We must check the left and right limits at this point:

• For \(x \geq 0\), as \(x\) approaches zero from the right, \(f(x) = x\). This means \(f(0) = 0\).
• For \(x < 0\), as \(x\) approaches zero from the left, \(f(x) = 3x\). Here, \(f(0) = 3(0) = 0\).

Both of these calculations show that \(f(x)\) approaches the value \(0\) from either side of the transition. Therefore, the function is continuous at \(x = 0\), with no jumps between the pieces at this point.

Analyzing Piecewise Functions

Analyzing piecewise functions involves systematically evaluating each part independently within its respective interval. Let's consider each segment in the given function, \(f(x)=\begin{cases} x & \text{for } x \geq 0 \ 3x & \text{for } x<0 \end{cases}\):

• **When \(x \geq 0\):** Here, \(f(x) = x\), which is a simple linear function with a slope of \(1\). This implies that the graph of this part will be a straight line passing through the origin and extending to positive infinity.
• **When \(x < 0\):** In this interval, \(f(x) = 3x\), also a linear function but with a steeper slope of \(3\). This line descends into the negative side of the graph, increasing (or decreasing more negatively) rapidly three times faster than the prior segment.
  

By combining this understanding with our analysis of intervals, one effectively maps the behavior of the piecewise function. Visualization helps confirm concepts such as continuity and overall shape behavior across each defined domain.