Problem 68 Sketch the graph of each piecewi... [FREE SOLUTION]

Chapter 3: Problem 68

Sketch the graph of each piecewise-defined function. Write the domain and range of each function. $$ f(x)=\left\\{\begin{aligned} x^{2} & \text { if } \quad x<0 \\ \sqrt{x} & \text { if } \quad x \geq 0 \end{aligned}\right. $$

Short Answer

Expert verified

The function is defined for all real numbers with a range of $ y \geq 0 $.

Step by step solution

Analyze the Function Parts

The given piecewise function is comprised of two parts. The first is $ f(x) = x^2 $ for $ x < 0 $, which is a parabola opening upwards. The second part is $ f(x) = \sqrt{x} $ for $ x \geq 0 $, representing the right half of a sideways parabola.

Determine the Domain

The domain is determined by looking at the values for which the function is defined. For $ x^2 $, the domain is $ x < 0 $, and for $ \sqrt{x} $, the domain is $ x \geq 0 $. Therefore, the overall domain of the function is all real numbers.

Find the Range of Each Part

For the part $ f(x) = x^2 $ where $ x < 0 $, the range is $ y > 0 $ because the parabola is above the x-axis for negative x-values. For the part $ f(x) = \sqrt{x} $ where $ x \geq 0 $, the range is $ y \geq 0 $ because the square root is defined for non-negative numbers and starts at $ y = 0 $. Together, these parts cover $ y \geq 0 $ for the entire function.

Sketch the Graph

Start with the parabola $ f(x) = x^2 $. Since $ x < 0 $, only the left half of this parabola is drawn, opening upwards starting from just left of the y-axis. Then sketch $ f(x) = \sqrt{x} $ for $ x \geq 0 $, starting from the origin (0,0) and increasing slowly to the right.

Verify Continuity at the Junction

The junction is at $ x = 0 $. At this point, the value from the function $ \sqrt{x} $ equals 0, while $ x^2 $ for values near 0 would be small positive numbers. The function is continuous at $ x = 0 $ as both parts meet here without gap or jump.

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Key Concepts

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

Graph Sketching of Piecewise Functions

When sketching the graph of a piecewise function, it is important to treat each piece individually while considering how they come together. In the function given, there are two distinct parts:

For the interval where $ x < 0 $, the function is described by $ f(x) = x^2 $. This represents a part of the standard parabola opening upwards. However, since $ x < 0 $ limits us to using only the left side of this parabola, we sketch half of it up to but not including the y-axis.
For $ x \geq 0 $, the function is $ f(x) = \sqrt{x} $. This is the graph of a square root function, starting from the origin and gradually increasing as $ x $ becomes larger.

To blend these graphs into a single sketch, align them at the point where $ x = 0 $. The transition should appear smooth, ensuring that they meet seamlessly at this junction point.

Domain and Range of Piecewise Functions

Understanding the domain and range of piecewise functions requires analyzing each piece separately. The domain encompasses all x-values that the function can legitimately accept:

For $ f(x) = x^2 $, $ x < 0 $, meaning this part of the function exists only to the left of the y-axis. Its domain is $ (-\infty, 0) $.
For $ f(x) = \sqrt{x} $, $ x \geq 0 $, indicating that it applies to x-values from zero onwards. Hence, this segment's domain is $ [0, \infty) $.
By combining these, the overall domain of the function is $ (-\infty, 0) \cup [0, \infty) $, equating to all real numbers.

The range looks at all possible y-values the function can output:

The range from $ f(x) = x^2 $ where $ x < 0 $ is strictly positive values, $ (0, \infty) $, because $ x^2 $ is always positive for any non-zero x.
The range from $ f(x) = \sqrt{x} $ where $ x \geq 0 $ is $ [0, \infty) $. This square root part allows y to be zero or any positive number, starting from zero and increasing.
Thus, the total range for the complete function is $ [0, \infty) $.

Function Continuity at Junctions

A critical aspect of piecewise functions is checking for continuity at the points where the individual pieces meet. In our case, this occurs at $ x = 0 $. To verify continuity, the function's value should converge consistently from both sides:

As $ x $ approaches 0 from the left, employing $ f(x) = x^2 $, the y-values become infinitesimally close to 0. Thus, there is a tendency towards the origin (0,0).
From the right, with $ f(x) = \sqrt{x} $ applied, the value at exactly $ x = 0 $ is 0. Therefore, the function does not exhibit any break or jump at that point.

These observations confirm that the function is continuous at the juncture $ x = 0 $. Each part aligns perfectly at this boundary, ensuring there's no abrupt change or discontinuity.

91影视

Sketch the graph of each piecewise-defined function. Write the domain and range of each function. $$ f(x)=\left\\{\begin{aligned} x^{2} & \text { if } \quad x<0 \\ \sqrt{x} & \text { if } \quad x \geq 0 \end{aligned}\right. $$

Short Answer

Step by step solution

Analyze the Function Parts

Determine the Domain

Find the Range of Each Part

Sketch the Graph

Verify Continuity at the Junction

Key Concepts

Graph Sketching of Piecewise Functions

Domain and Range of Piecewise Functions

Function Continuity at Junctions

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