Problem 74 - College Algebra with Modeling and Visualization

Question

Exercises \(69-74:\) Complete the following for \(f(x)\) (a) Determine the domain of \(f\) (b) Evaluate \(f(-2), f(0),\) and \(f(3)\) (c) Graph \(f\) (d) Is \(f\) continuous on its domain? $$ f(x)=\left\\{\begin{array}{ll} 3 & \text { if }-4 \leq x \leq-1 \\ x-2 & \text { if }-1

Step-by-Step Solution

Answer

Domain: \([-4, 4]\); \(f(-2) = 3\), \(f(0) = -2\), \(f(3) = 1.5\); Discontinuous on its domain.

Step 1: Determine the Domain of f

The function \( f(x) \) is defined piecewise with different expressions over different intervals of \( x \). We analyze each range:\1. \( -4 \leq x \leq -1 \) 2. \( -1 < x \leq 2 \) 3. \( 2 < x \leq 4 \)By combining these intervals, we can express the domain of \( f \) as \( [-4, -1] \cup (-1, 2] \cup (2, 4] \). Therefore, the domain of \( f \) is \([-4, 4]\).

Step 2: Evaluate f(x) at Specific Points

We need to evaluate \( f(x) \) at \( x = -2, 0, \) and \( 3 \). For each value:- \( f(-2) \): Since \( -4 \leq -2 \leq -1 \), use the first expression, so \( f(-2) = 3 \).- \( f(0) \): Since \( -1 < 0 \leq 2 \), use the second expression, \( f(0) = 0 - 2 = -2 \).- \( f(3) \): Since \( 2 < 3 \leq 4 \), use the third expression, \( f(3) = 0.5 \times 3 = 1.5 \).

Step 3: Graph the Function

Graph each piece of the piecewise function:- For \( -4 \leq x \leq -1 \), draw a horizontal line at \( y = 3 \).- For \( -1 < x \leq 2 \), plot the line \( y = x - 2 \), starting just above \( x = -1 \) and ending at \( x = 2 \).- For \( 2 < x \leq 4 \), plot the line \( y = 0.5x \), starting just above \( x=2 \) and ending at \( x=4 \). Use open or closed circles at endpoints to indicate inclusivity based on the intervals.

Step 4: Check Continuity

A function is continuous on its domain if there are no breaks, jumps, or holes.- At \( x = -1 \): Left of \( -1 \) it is 3, right of \( -1 \) it is slightly above 1. There is a jump.- At \( x = 2 \): Left of 2 \( f(x) \to 0 \), right of 2 \( f(x) \to 1 \). There is a jump.Thus, \( f \) is not continuous on its domain due to jumps at \( x = -1 \) and \( x = 2 \).

Key Concepts

Understanding the Domain of Functions

The domain of a function is essentially the set of all input values (usually represented by \( x \)) that the function can accept without any issues. For piecewise functions like the one in the exercise, the function is defined by different expressions over specific intervals. In this case, the function \( f(x) \) has three distinct pieces:

• \( -4 \leq x \leq -1 \)
• \( -1 < x \leq 2 \)
• \( 2 < x \leq 4 \)

Each piece applies only to a certain range of \( x \). By combining these intervals, we can see that any \( x \) value between -4 and 4 fits into one of the interval pieces. Thus, the domain of \( f(x) \) is all real numbers from -4 to 4, expressed in interval notation as \([-4, 4]\).

Understanding the domain helps us define where the function exists and prevents us from inputting values that could "break" the function, which is especially crucial in piecewise functions.

Function Evaluation Made Simple

Evaluating a function at a given point means finding its output value based on the input value. For this piecewise function, we determine which part (or piece) of the function to use based on the input value's interval. Here's how to evaluate \( f(x) \) at \( x = -2, 0, \) and \( 3 \):

• For \( f(-2) \): Look at where \(-2\) falls, which is in the first interval, so \( f(-2) = 3 \).
• For \( f(0) \): This is within the second interval, hence \( f(0) = 0 - 2 = -2 \).
• For \( f(3) \): \( 3 \) is in the third interval, so the calculation is \( f(3) = 0.5 \times 3 = 1.5 \).

Remember, the key is to find which interval the input value belongs to and then use the corresponding rule to calculate the output.

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each piece according to its rules over the specified interval. This can be a bit like connecting the dots, but with guidelines for each section of the function.

For the given function \( f(x) \):

• From \(-4\) to \(-1\), plot a horizontal line at \( y = 3 \). This represents the first piece of the function.
• From \(-1\) just up to 2, plot the line \( y = x - 2 \). This line will fall between the points on the left and jump slightly on the right at 2.
• From a little over 2 to 4, plot the line \( y = 0.5x \), indicating the linear increase in this last section.

Being careful with endpoints matters; use closed dots for included values and open dots where values are excluded to demonstrate precise domain inclusivity at those points.

Exploring Continuity of Functions

Continuity is an important feature when studying functions, especially in calculus. A function is continuous at a point when there are no breaks, jumps, or holes in the graph at that point. For the function \( f(x) \), checking continuity involves examining the transition points between different pieces.

Let's look at the critical points:

• At \( x = -1 \): The function jumps from one piece to another, from \( y = 3 \) to somewhere slightly above 1. This jump indicates discontinuity.
• At \( x = 2 \): The function has a similar issue, transitioning from a lower value to a higher one, marking another jump.

Due to these jumps at \( x = -1 \) and \( x = 2 \), \( f(x) \) is not continuous across its whole domain. Understanding continuity in functions helps predict behavior and identify where problems may arise.