Problem 77 - College Algebra with Modeling and Visualization

Question

Use \(f(x)\) to complete the following: $$ f(x)=\left\\{\begin{array}{ll} 3 x-1 & \text { if }-5 \leq x<1 \\ 4 & \text { if } 1 \leq x \leq 3 \\ 6-x & \text { if } \quad 3

Step-by-Step Solution

Answer

Evaluations: \(f(-3) = -10\), \(f(1) = 4\), \(f(2) = 4\), \(f(5) = 1\). Constancy: \([1, 3]\). \(f\) is discontinuous.

Step 1: Evaluate f at x = -3

The function applicable for \(x = -3\) is \(f(x) = 3x - 1\) for \(-5 \leq x < 1\). Substitute \(x = -3\) into the equation: \[ f(-3) = 3(-3) - 1 = -9 - 1 = -10 \] So, \(f(-3) = -10\).

Step 2: Evaluate f at x = 1

The function applicable for \(x = 1\) is the constant function \(f(x) = 4\) for \(1 \leq x \leq 3\). Therefore, \[ f(1) = 4 \]

Step 3: Evaluate f at x = 2

Similarly, for \(x = 2\), which lies within the interval \(1 \leq x \leq 3\), the function value remains constant as \[ f(2) = 4 \]

Step 4: Evaluate f at x = 5

The function applicable for \(x = 5\) is \(f(x) = 6 - x\) for \(3 < x \leq 5\). Substitute \(x = 5\) into the equation: \[ f(5) = 6 - 5 = 1 \]

Step 5: Determine the interval where f is constant

The function \(f(x)\) is constant equal to 4 over the interval from \(1\) to \(3\), inclusive. Thus, the interval is \([1, 3]\).

Step 6: Sketch the graph of f

The graph of \(f(x)\) consists of three segments: 1. A line segment from \((-5, -16)\) to \((1, 2)\) using the equation \(3x - 1\).2. A horizontal line from \((1, 4)\) to \((3, 4)\) where \(f(x) = 4\).3. A line segment from \((3, 3)\) to \((5, 1)\) using the equation \(6 - x\).The graph is not continuous because there is a jump discontinuity at \(x = 3\).

Step 7: Assess the continuity of f

The function \(f(x)\) is not continuous across its domain due to a jump from \(f(x) = 4\) at \(x = 3\) to \(f(x) = 3\) when approaching from the right. This indicates a discontinuity at \(x = 3\).

Key Concepts

Function Evaluation

Function evaluation means finding the output of a function given an input value. For piecewise functions, it's essential to first determine which part of the function to use, based on the input value's interval. Here's how you can evaluate such functions easily:

• Identify the interval in which the given value of x lies.
• Use the corresponding equation or rule attached to that interval to find the function's value.
• Solve the expression to obtain the result.

For example, our function is defined in different parts. Evaluating at different x values, we proceed as follows:- For \( x = -3 \), since \(-5 \leq x < 1\) applies, use the first rule: \(f(x) = 3x - 1\). Hence, \( f(-3) = -10 \).- For \( x = 1 \) and \( x = 2 \), both fall in the interval \(1 \leq x \leq 3\), where \(f(x) = 4\). Therefore, \( f(1) = 4 \) and \( f(2) = 4 \).- Finally, \( x = 5 \) is within \(3 < x \leq 5\), so use \(f(x) = 6 - x\). Here, \( f(5) = 1 \).

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each segment independently and then combining them on the same graph. For our function, more steps are involved:

• Plot each component over its specific interval.
• Connect the plotted points to form the respective segments; lines or curves, as applicable.
• Use open or closed circles to indicate whether endpoints are included or excluded.

For instance, - The segment from \( -5 \leq x < 1 \) uses \(f(x) = 3x - 1\). Draw a line from \((-5, -16)\) to just before \((1, 2)\) with an open circle.- From \( 1 \leq x \leq 3 \), draw a horizontal line at \(y = 4\) from \((1, 4)\) to \((3, 4)\). Both ends are closed since the interval is fully inclusive.- From \( 3 < x \leq 5 \), use \(f(x) = 6 - x\). Draw from just above \((3, 3)\) to \((5, 1)\), with the latter being closed.This creates a graph composed of distinct segments, reflecting the piecewise nature.

Continuity in Functions

Function continuity means there are no breaks, jumps, or holes in its graph—it flows smoothly over its domain. A function is continuous if, at every point, the function's value matches its limiting behavior from both sides. For piecewise functions, this can be trickier, as each segment may or may not connect smoothly.When analyzing piecewise functions for continuity:

• Examine border points between intervals closely.
• Check if the function values and limits match at these points from both sides.

In our specific example, at \(x = 3\):- From the left, \(f(x)\) is constant at 4. Approaching from the right, however, it's defined by \(6 - x\), which evaluates to 3.- Since there is a difference between these values, a jump discontinuity occurs.As such, the function is not continuous at \(x = 3\), reflecting a key characteristic that needs to be considered when dealing with piecewise functions. This illustrates how closely tied continuity is with how pieces are connected on the graph.