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Chapter 2: Problem 41

Sketch a 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 function is \(f(x)=2\) for \(x\leq-1\) and \(f(x)=x^2\) for \(x>-1\).

Step by step solution

01

Analyze the Function Definition

The function is defined as a piecewise function with two parts. For \(x\leq -1\), the function is \(f(x) = 2\), which is a constant function. For \(x > -1\), the function is \(f(x) = x^2\), which is a quadratic function (a parabola) that opens upwards.
02

Identify the Domain of Each Piece

The first piece \(f(x) = 2\) is defined for \(x \leq -1\). This means it includes all values of \(x\) from \(-\infty\) to \(-1\) including \(-1\). The second piece \(f(x) = x^2\) is defined for \(x > -1\), which includes all values of \(x\) strictly greater than \(-1\).

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Key Concepts

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

Graphing Functions
Graphing functions involves understanding the visual representation of mathematical equations. It is a crucial skill in mathematics that helps us see the relationship between variables. For piecewise functions, it's essential to consider each piece of the function separately:

  • Constant Function: In our example, for all values of \(x\) less than or equal to \(-1\), \(f(x) = 2\). This part appears as a horizontal line at \(y = 2\) from \(-\infty\) to \(-1\), making a closed dot at \(x = -1\) indicating inclusion.
  • Quadratic Function: For \(x > -1\), the function \(f(x) = x^2\) forms a curve (called a parabola) that opens upward. Start plotting just right of \(x = -1\), with an open dot at \(x = -1\) indicating exclusion. Continue drawing the parabola more to the right.
Bringing both parts together creates the complete graph of the piecewise function. The transition point at \(x = -1\) is critical since it dictates where and how the graph changes.
Function Analysis
Function analysis allows you to deeply understand the behavior and characteristics of a function. For a piecewise function:

  • Domain: This refers to all possible input values. Here, the domain for our function is \(-\infty < x < \infty\), since no \(x\) is restricted.
  • Range: This represents all possible outputs. For our function, the range combines outputs from both pieces: For the constant piece, the output is always \(2\). For the quadratic piece, outputs start slightly above \(0\) (considering it's open at \(0\)) and extend upwards towards infinity.
  • Continuity: A function is continuous if there are no breaks or jumps when moving from one section to another. Our function has a jump at \(x = -1\) since it suddenly jumps from 2 to just above 0.
Analyzing these details offers comprehensive insights into how the function behaves across its entire domain.
Quadratic Functions
Quadratic functions are an integral part of studying algebra. A standard quadratic function is represented as \(f(x) = ax^2 + bx + c\):

  • Shape and Open Direction: The graph of a quadratic function is a parabola. Here, \(f(x) = x^2\) means it opens upward because the coefficient of \(x^2\) is positive.
  • Vertex: Typically, the vertex is the highest or lowest point of the parabola. In \(f(x) = x^2\), the vertex is at the origin \((0,0)\), indicating it's the lowest point with symmetry around the y-axis.
  • Behavior at Infinity: As \(x\) moves towards either positive or negative infinity, the value of \(x^2\) increases, causing the parabola to rise indefinitely.
Understanding quadratic functions and their properties help to evaluate and predict their behavior effectively, which is especially useful for graphing piecewise functions like in our exercise.

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Most popular questions from this chapter

Sketch a graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll} 0 & \text { if }|x| \leq 2 \\ 3 & \text { if }|x|>2 \end{array}\right.$$

The power produced by a wind turbine depends on the speed of the wind. If a windmill has blades 3 meters long, then the power \(P\) produced by the turbine is modeled by $$P(v)=14.1 v^{3}$$ where \(P\) is measured in watts (W) and \(v\) is measured in meters per second (m/s). Graph the function \(P\) for wind speeds between \(1 \mathrm{m} / \mathrm{s}\) and \(10 \mathrm{m} / \mathrm{s}\). (IMAGE CAN'T COPY).

For every integer \(n\), the graph of the equation \(y=x^{n}\) is the graph of a function, namely \(f(x)=x^{n} .\) Explain why the graph of \(x=y^{2}\) is not the graph of a function of \(x .\) Is the graph of \(x=y^{3}\) the graph of a function of \(x ?\) If so, of what function of \(x\) is it the graph? Determine for what integers \(n\) the graph of \(x=y^{n}\) is a graph of a function of \(x\)

Graphing Functions Sketch a graph of the function by first making a table of values. $$g(x)=(x-1)^{3}$$

Sketch a graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll} 1-x & \text { if } x<-2 \\ 5 & \text { if } x \geq-2 \end{array}\right.$$
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