Problem 44 Sketch the graph of the piecewis... [FREE SOLUTION]

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Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 2: Problem 44

Sketch the graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll}x^{2} & \text { if }|x| \leq 1 \\\1 & \text { if }|x|>1\end{array}\right.$$

Short Answer

Expert verified

The graph consists of a parabola $ f(x) = x^2 $ from $[-1, 1]$ and horizontal lines $ f(x) = 1 $ elsewhere.

Step by step solution

Identify the function components

The piecewise function has different rules depending on the value of $ x $. The function $ f(x) = x^2 $ applies for $ |x| \leq 1 $, which means $ x $ is between $-1$ and $1$ inclusive. The constant function $ f(x) = 1 $ applies for $ |x| > 1 $.

Sketch $ f(x) = x^2 $ for $ |x| \leq 1 $

For $ x $ values between $-1$ and $1$, plot the parabola $ f(x) = x^2 $. At $ x = -1 $, the graph is at $ f(-1) = (-1)^2 = 1 $, at $ x = 0 $, $ f(0) = 0^2 = 0 $, and at $ x = 1 $, $ f(1) = 1^2 = 1 $. These points form a curve that is a section of the parabola.

Sketch $ f(x) = 1 $ for $ |x| > 1 $

For $ x $ less than $-1$ and greater than $1$, $ f(x) = 1 $. Thus, draw a horizontal line at $ y = 1 $. The line extends outward from $-\infty$ to $-1$ and from $1$ to $\infty$.

Combine the segments and check continuity

Combine the sketch of the parabola and the horizontal lines. The function is continuous at $ x = -1 $ and $ x = 1 $ since $ x^2 = 1 $ and both pieces join at these points smoothly. Therefore, the graph should look like a parabola restricted to the interval $[-1, 1]$ and flat horizontal segments at $ y = 1 $ outside this interval.

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Key Concepts

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

Graphing Piecewise Functions

Graphing piecewise functions involves plotting different portions of a function over specified intervals. It requires understanding how each segment behaves in its domain. For this particular problem, the function is broken into two segments: a curve and a line.

For $f(x) = x^2$ where $|x| \leq 1$, a parabola defines the function. This is visualized by plotting points: at $x = -1$ and $x = 1$, $f(x) = 1$. At $x = 0$, $f(x) = 0$.
Outside this range, when $|x| > 1$, $f(x) = 1$. This results in a horizontal line for values less than \-1\ otherwise greater than \1\.

By plotting each segment within its domain, you create a complete picture of the function. It helps to draw these segments on the coordinate plane and observe how they interact. This process brings clarity in understanding the behavior of piecewise-defined functions.

Continuity of Piecewise Functions

The concept of continuity in piecewise functions examines whether the function remains uninterrupted across segments. For a function to be continuous, its graph must not have jumps or breaks at the domain transition points.

In this exercise, continuity is checked at the endpoints $x = -1$ and $x = 1$, where the function definition changes. Here, $f(x) = x^2$ connects seamlessly with $f(x) = 1$, sustaining the same value at these points.

The continuity can be confirmed by checking:

At $x = -1$ and $x = 1$, the parabolic segment leads to $f(x) = 1$. Hence, the horizontal line also reaches $y = 1$, indicating continuity at the junction.
Transition involves no sudden jumps, ensuring a smooth connection throughout.

Thus, effectively managing the transitions from one piece to another ensures the entire function is continuous, delivering a smooth graph across its domain.

Parabolas in Piecewise Functions

Parabolas often feature in piecewise functions as they represent quadratic relationships. Understanding their shape is key when graphing pieces of these functions. A parabola's basic form, such as $f(x) = x^2$, is a symmetric curve that opens upwards starting from the vertex at the origin.

When used in piecewise functions, the parabola must fit within its defined segment. For this exercise:

The segment where $|x| \leq 1$ forms a section of $f(x) = x^2$. Here, the parabola begins at $-1$, rises to its vertex at $0$, and descends again through $1$.
It is limited to this interval, appearing as just a part of the full quadratic curve.

Grasping how parabolas behave aids in piecing together the segment and ensures precise placement within the larger function. Such clarity in understanding allows for their accurate representation in piecewise graphs.

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91影视

Sketch the graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll}x^{2} & \text { if }|x| \leq 1 \\\1 & \text { if }|x|>1\end{array}\right.$$

Short Answer

Step by step solution

Identify the function components

Sketch \( f(x) = x^2 \) for \( |x| \leq 1 \)

Sketch \( f(x) = 1 \) for \( |x| > 1 \)

Combine the segments and check continuity

Key Concepts

Graphing Piecewise Functions

Continuity of Piecewise Functions

Parabolas in Piecewise Functions

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