Problem 73 In Exercises $53-76$, graph th... [FREE SOLUTION]

Chapter 1: Problem 73

In Exercises $53-76$, graph the piccewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant. $$f(x)=\left\\{\begin{array}{ll} x+3 & x \leq-2 \\ |x| & -2< x< 2 \\ x^{2} & x \geq 2 \end{array}\right.$$

Short Answer

Expert verified

Domain: $(-b, b)$, Range: $(-b, 1] \cup [0, 2) \cup [4, b)$. \nDecreasing on $(-b, -2) \cup (-2, 0)$, Increasing on $(0, 2) \cup [2, b)$.

Step by step solution

Analyze Domain

The function is defined in three distinct parts. Examine each piece to determine its domain segment:\[ f(x) = \begin{cases} x+3 & x \leq -2 \ |x| & -2 < x < 2 \ x^2 & x \geq 2 \end{cases} \]- The domain of the first piece $ x+3 $ is $(-b, -2]$.- The domain of the second piece $ |x| $ is $(-2, 2)$. - The domain of the third piece $ x^2 $ is $[2, b)$. Combining these, the function's domain is $(-b, b)$.

Analyze Range

Determine the values that the function can take, given each segment:- For $ x+3 $, the outputs depend on $ x \leq -2 $. Calculating for $ x=-2 $, we get $ x+3 = 1 $ and it continues decreasing as $ x $ decreases, so the range is $(-b, 1]$.- For $ |x| $, the function values are from 0 up to numbers less than 2. Thus, this contributes $[0, 2)$ to the range.- For $ x^2 $, starting at $ x=2 $, the value is 4 and increases as $ x $ increases, giving a range of $[4, b)$.Combine these, the overall range is $(-b, 1] \cup [0, 2) \cup [4, b)$.

Intervals of Increase, Decrease, and Constant Specific Segments

Determine increasing, decreasing, and constant intervals for each segment:- $ x+3 $ is a linear function decreasing for $ x \leq -2 $ as the slope is negative.- $ |x| $ is a V-shaped function. - Decreasing: $(-2, 0)$ because it's descending towards 0. - Increasing: $(0, 2)$ as it rises away from 0.- $ x^2 $ is a quadratic function that increases for $ x \geq 2 $ as it concaves up.Combining, the function is:- Decreasing on $(-b, -2) \cup (-2, 0)$.- Increasing on $(0, 2) \cup [2, b)$.- There are no constant intervals.

Graphing the Function

Create a graph by plotting each piece separately and highlighting transitions:- For $ x+3 $, graph starts from $ x=-2 $ down to as $ x $ tends towards $-b$.- For $ |x| $, plots between $-2 < x < 2$, getting the V-shape, with minimum at $ x=0 $.- For $ x^2 $, start at $ (2, 4) $ and then concave upwards.Use open and closed circles to denote where pieces start and end to ensure continuity where specified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Function Domain

In mathematics, the domain of a function is the complete set of possible values of the independent variable, usually denoted as $ x $. For piecewise functions, the domain is determined by considering all the segments that make up the function.
Let's break it down for the given piecewise function:

For the segment $ f(x) = x + 3 $, the domain is $(-\infty, -2]$. This covers all $ x $ values less than or equal to -2.
For the segment $ f(x) = |x| $, the domain is $(-2, 2)$. This only includes $ x $ values between -2 and 2.
For the segment $ f(x) = x^2 $, the domain is $[2, \infty)$. This entails all $ x $ values greater than or equal to 2.

Combining these segments, we find that the total domain of the piecewise function is $(-\infty, \infty)$. This means any real number can be plugged into the function, as each point is accounted for within at least one of the pieces.

Exploring the Function Range

The range of a function is the set of all possible output values (dependent variable, usually $ y $). For piecewise functions, it helps to look at each segment separately:

For $ f(x) = x + 3 $ when $ x \leq -2 $, the smallest output starts at 1 and decreases without bound as $ x $ goes to $-\infty$. Therefore, the range for this part is $(-\infty, 1]$.
For $ f(x) = |x| $ within $-2 < x < 2$, the outputs start from slightly above 0 and rise just below 2. Thus, the range here is $[0, 2)$.
For $ f(x) = x^2 $ where $ x \geq 2 $, the outputs begin at 4 and increase indefinitely as $ x $ increases. This gives a range of $[4, \infty)$.

Together, the entire function's range is $(-\infty, 1] \cup [0, 2) \cup [4, \infty)$, covering output values produced across all segments.

Identifying Increasing and Decreasing Intervals

A function's behavior in terms of increasing and decreasing intervals indicates where its output values grow or shrink as $ x $ moves through the domain. By analyzing each piece:

The function $ f(x) = x + 3 $ decreases for $ x \leq -2 $. The slope is negative, meaning as $ x $ increases, $ f(x) $ decreases.
In $ f(x) = |x| $, there is a notable V-shaped behavior:
- Between $-2 $ and 0, the function decreases as $ x $ approaches 0.
- Between 0 and 2, the function increases as $ x $ moves away from 0.
For $ f(x) = x^2 $, the function is increasing when $ x \geq 2 $ due to its upward concave shape.

In summary, the function decreases over $(-\infty, -2) \cup (-2, 0)$ and increases over $(0, 2) \cup [2, \infty)$. There are no constant intervals due to the nature of the equations involved.

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each segment according to its specific domain and observing the transitions between these segments. Here's how to graph the given piecewise function:

Start by graphing $ f(x) = x + 3 $ for $ x \leq -2 $, noting a downward slope. This segment starts with a closed circle at $ x = -2 $.
Next, plot $ f(x) = |x| $ for $-2 < x < 2$. This forms a V-shape, starting from a point just above 0, increasing as $ x $ moves towards 2.
Finally, graph $ f(x) = x^2 $ for $ x \geq 2 $. Start at the point (2, 4) and show an upward curve, typical of a parabola.

When graphing, use open circles to denote values excluded in the function (e.g., $ x = -2 $ for the second segment from the left) and closed circles where values are included. This helps in visualizing the continuity or discontinuity at transitions between segments.

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Short Answer

Step by step solution

Analyze Domain

Analyze Range

Intervals of Increase, Decrease, and Constant Specific Segments

Graphing the Function

Key Concepts

Understanding the Function Domain

Exploring the Function Range

Identifying Increasing and Decreasing Intervals

Graphing Piecewise Functions

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