Problem 69 Sketch the graph of each piecewi... [FREE SOLUTION]

Chapter 3: Problem 69

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

Short Answer

Expert verified

Domain: $(-\infty, \infty)$; Range: $(-\infty, 2)$.

Step by step solution

Identify the Conditions

The function is piecewise-defined, meaning it behaves differently based on the value of $x$. The first part $|x-2|$ applies when $x < 0$, and the second part $-x^2$ applies when $x \geq 0$.

Analyze Each Piece

For $x < 0$, the expression is $|x-2|$. Since $x < 0$, $x - 2$ is negative, thus $|x-2| = 2-x$. For $x \geq 0$, the expression is $-x^2$, a downward-opening parabola.

Determine the Domain

The domain of $g(x)$ is determined by the union of the intervals defined by the piecewise function. Hence, the domain is $ (-\infty, \infty) $ as both intervals cover all real numbers together.

Determine the Range

The first piece $2-x$ for $x < 0$ increases without bound as $x$ approaches 0 from the left, giving range $(2, \infty)$. The second piece $-x^2$ for $x \geq 0$ yields the range $(-\infty, 0]$. Combining these, the range of $g(x)$ is $(-\infty, 2)$.

Sketch the Graph

On the left of the y-axis ($x < 0$), graph the line from $f(x) = 2-x$. Start from the point where $x \to -\infty$ and approach the point at $x = 0$, open circle at $(0,2)$. On the right ($x \geq 0$), graph the parabola $-x^2$, which touches the y-axis at 0 going downward.

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Key Concepts

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

Domain and Range

Understanding the domain and range of piecewise-defined functions is crucial for analyzing them. The **domain** refers to all possible input values (x-values) for the function. In our example, the domain is the set of all real numbers, [ (-鈭�, 鈭�), because each piece of the function is defined either for all x less than 0 or all x greater than or equal to 0.
The **range** of a function is the set of all possible output values. For our piecewise-defined function, we have distinct ranges for each piece:

For x < 0, the absolute value function |x-2| results in values ranging from 2 to 鈭� (as x approaches 0 from the left), yielding (2, 鈭�).
For x 鈮� 0, the quadratic function -x² produces outputs from -鈭� to a maximum of 0 (at x=0), resulting in (-鈭�, 0].

This leads us to the overall range of the function being (-鈭�, 2), as the two pieces hold different intervals for their outputs.

Graphing Functions

To graph piecewise-defined functions, we plot each segment separately. This involves carefully considering where each portion of the function applies, using open or closed circles to indicate whether endpoints are included.

For the segment where x < 0, the function is defined by |x-2| = 2-x. Since |x-2| starts with an open circle at (0,2), we draw this linear segment as a straight line extending leftwards without any upper boundary.
For x 鈮� 0, we plot the parabola -x². This quadratic segment touches the y-axis at the point (0,0), where the function changes, and it opens downward, curving downwards.

The accurate depiction of each piece and understanding where each part starts and stops is critical. Arrows are used to show the continuing direction of lines or curves, helping to visualize the entire graph clearly.

Absolute Value

The **absolute value** of a number is its distance from zero on the number line, disregarding its sign. For the function |x-2| when x<0, considering x - 2 makes it clearer. Since x is always less than zero in this part, x - 2 will be negative, requiring you to "flip" the sign inside the absolute value bars to 2-x.
This subtle flipping is why |x-2| transforms into 2-x when x is negative. The absolute value function typically results in a V-shaped graph; however, due to the conditions given, only the linear path for x < 0 is seen here.

Quadratic Functions

Quadratic functions are characterized by the variable x being squared, which forms a parabolic shape when graphed. The general form is ax虏 + bx + c. For the piece -x², we specifically have a downward-opening parabola since the coefficient of x² is negative.
In this case, given the condition x 鈮� 0, only the right side of the parabola is visible. It starts at the origin and curves downward because of the -x² terms. Key characteristics of quadratic functions include:

Vertex: The highest or lowest point of the parabola. Here, it's a maximum at (0,0).
Axis of symmetry: A vertical line that divides the parabola into symmetrical halves. Here, it is the y-axis itself.
Direction: The parabola opens downward, as indicated by the negative leading coefficient.

Recognizing the structure and features of quadratic functions is essential for interpreting their graphs accurately.

91影视

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

Short Answer

Step by step solution

Identify the Conditions

Analyze Each Piece

Determine the Domain

Determine the Range

Sketch the Graph

Key Concepts

Domain and Range

Graphing Functions

Absolute Value

Quadratic Functions

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