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Chapter 2: Problem 41

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

Short Answer

Expert verified
The graph consists of a horizontal line at \(y = 2\) for \(x \leq -1\) and a parabola \(y = x^2\) for \(x > -1\).

Step by step solution

01

Understanding the Piecewise Function

This function is presented in two distinct parts: when \(x \leq -1\), the function value is a constant \(f(x) = 2\), and when \(x > -1\), the function becomes \(f(x) = x^2\). These two functional forms need to be interpreted in their respective domain conditions.
02

Graphing for \(x \leq -1\)

When \(x \leq -1\), the function is constant and equal to 2. This means for any \(x\) that is less than or equal to \(-1\), the graph is a horizontal line at \(y = 2\). Specifically, this line extends from all negative values of \(x\) up to \(-1\) and includes the point \((-1, 2)\).
03

Graphing for \(x > -1\)

When \(x > -1\), the function is a parabola given by \(f(x) = x^2\). This part of the graph starts just to the right of \(x = -1\) and follows the shape of \(x^2\), a typical parabolic curve opening upwards with its vertex at the origin \((0,0)\), but starting only at \(x = -1\).
04

Connecting the Two Parts

For the value \(x = -1\), the function value according to the first piece is \(f(-1) = 2\), forming an endpoint for the left segment. For values greater than \(-1\), the curve \(y = x^2\) starts, moving upwards away from the point \((-1, 1)\). In the graph, the point at \((-1, 1)\) for the parabola is open, indicating it's not included in the graph of \(f(x)\) due to the condition \(x > -1\). Thus, the graph seamlessly transitions from a line to a curve at \(x = -1\).

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Key Concepts

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

Graphing Piecewise Functions
Piecewise functions can be thought of as a combination of multiple functions, each defined over a specific part of the domain. The given function \( f(x) = \left\{\begin{array}{ll}2 & \text { if } x \leq-1 \x^{2} & \text { if } x>-1\end{array}\right. \) demonstrates this beautifully. Here, the function is defined by two separate expressions:
  • The first case, \( f(x) = 2 \) for \( x \leq -1 \), indicates that the function takes a constant value of 2 over this range.
  • The second case, \( f(x) = x^2 \) for \( x > -1 \), shows the function taking on a quadratic form, a typical parabola.
To sketch the graph of a piecewise function, identify boundaries where the function changes its form; here it's \( x = -1 \). Each section of the graph should be drawn according to its corresponding formula, respecting these boundaries. Understanding the nature of each segment individually helps in achieving a clear and accurate graph.
Continuous and Discontinuous Functions
It is essential to understand the concept of continuity when graphing piecewise functions. A function is said to be continuous if there are no sudden jumps, breaks, or holes in its graph.
In this exercise, the piecewise function includes both a constant segment and a quadratic segment:
  • For \( x \leq -1 \), the function is constant and continuous as it is expressed by \( f(x) = 2 \) from the left up to and including the point \( x = -1 \), which forms a closed endpoint \((-1, 2)\).
  • For \( x > -1 \), \( f(x) = x^2 \), starts right after \( x = -1 \) and is again continuous but does not include the endpoint \((-1, 1)\). It starts without including this open point. This means the function is not continuous at \( x = -1 \). The graph transitions from a horizontal line to a parabolic curve but with a tiny jump at this boundary.
The function is therefore continuous on each of its segments but has a discontinuity at \( x = -1 \), where the function value jumps.
Parabolic Curves
Parabolic curves are common in mathematics, often described by functions like \( y = x^2 \). They feature a "U"-shaped curve that can open upwards or downwards:
  • In \( f(x) = x^2 \), as is the case for \( x > -1 \) in our piecewise function, the parabola opens upwards, with its vertex typically located at the origin \((0, 0)\) if extended indefinitely.
  • For this problem, the parabola starts at \( x > -1 \). It reflects the continuous growth of the squared term, showing how the output increases swiftly as \( x \) increases.
The graph of a parabolic function can be illustrated more vividly by plotting points or using its vertex and symmetrical properties. Due to its symmetric nature, for any value of \( x \), \( f(x) \) and \( f(-x) \) would yield the same result if extended symmetrically, though in our piecewise function case, it only considers positive \( x \) values beyond \( -1 \). Analyzing the behavior of these curves is crucial for understanding their intersection, direction, and overall graph shape.

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