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Chapter 1: Problem 98

Graph the function \(f\) given by $$f(x)=\left\\{\begin{array}{ll} -3, & \text { for } x=-2 \\ x^{2}, & \text { for } x \neq-2 \end{array}\right.$$

Short Answer

Expert verified
Plot (-2, -3) and the parabola \(饾憮(饾懃) = 饾懃虏\) with an open circle at (-2, 4).

Step by step solution

01

- Understand the Function Definition

The function is defined piecewise. For the specific value of 饾懃 = -2, the function value is -3. For all other values of 饾懃, the function follows the equation 饾憮(饾懃) = 饾懃虏.
02

- Plot the Specific Value

At 饾懃 = -2, 饾憮(饾懃) = -3. Plot the point (-2, -3) on the graph.
03

- Plot the Parabola

For all other values of 饾懃, plot the function 饾憮(饾懃) = 饾懃虏. This is a parabola opening upwards. Calculate a few values such as: \(\begin{align*} \text{At} \, x = -3, & \, f(x) = 9 \ x = -1, & \, f(x) = 1 \ x = 0, & \, f(x) = 0 \ x = 1, & \, f(x) = 1 \ x = 2, & \, f(x) = 4 \ x = 3, & \, f(x) = 9 \text \endyt}\). Plot these points and sketch the parabola.
04

- Consider Discontinuity

Note that the function is discontinuous at \(x = -2\). The point (-2, 4) is not part of the graph of 饾憮(饾懃). Here's how it looks: Along the curve of 饾憮(饾懃) = 饾懃虏, put an open circle at (-2, 4) to indicate that 饾憮(饾懃) is not defined there for this part of the graph.
05

- Final Graph

Combine the parabola with the special defined point to complete the graph. The graph is the parabola 饾憮(饾懃) = 饾懃虏 with an open circle at (-2, 4), and in addition, the single point (-2, -3) representing the specific value at 饾懃 = -2.

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Key Concepts

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

headline of the respective core concept
Piecewise functions can sometimes seem tricky, but they become manageable when broken down step by step. The function given in the exercise is an example of a piecewise function, meaning it has different rules for different parts of its domain.
Here, the function has a specific value at 饾懃 = -2 and follows a different rule for all other values of 饾懃. This creates an interesting scenario on the graph involving a point and a curve, making it crucial to understand both aspects. Let's delve deeper into the main concepts involved.
headline of the respective core concept
A key part of understanding this exercise is analyzing the parabola involved in the function. For any value of 饾懃 other than -2, the function simplifies to 饾憮(饾懃) = 饾懃虏. This happens to be the equation of a parabola that opens upwards.
A parabola is a symmetrical plane curve. It has a vertex (the lowest or highest point) and it opens either upwards or downwards depending on the coefficient of the quadratic term. In this exercise, since the coefficient of 饾懃虏 is positive, the parabola opens upwards.
To graph the parabola, you can calculate values of 饾憮(饾懃) for several different 饾懃 values. For example:
  • At 饾懃 = -3, 饾憮(饾懃) = 9
  • At 饾懃 = -1, 饾憮(饾懃) = 1
  • At 饾懃 = 0, 饾憮(饾懃) = 0
  • At 饾懃 = 1, 饾憮(饾懃) = 1
  • At 饾懃 = 2, 饾憮(饾懃) = 4
  • At 饾懃 = 3, 饾憮(饾懃) = 9
Plot these points on a graph, and sketch the smooth curve of the parabola passing through them. Remember, the graph is a visual representation of all possible points that satisfy the equation 饾憮(饾懃) = 饾懃虏.
headline of the respective core concept
Discontinuity is another critical concept in this exercise. A discontinuity occurs where a function is not continuous, meaning there's a

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