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Chapter 1: Problem 2

State the range for the given functions. Graph each function. $$ f(x)=x^{2}, x \in[0,2] $$

Short Answer

Expert verified
The range of the function is \([0, 4]\).

Step by step solution

01

Understand the Function

The function given is a quadratic function, specifically a parabola opening upwards, since the term with the highest power, \(x^2\), has a positive coefficient. This is a fundamental characteristic of quadratic functions.
02

Identify the Domain

The function is defined for \(x\) within the closed interval \([0, 2]\). This means \(x\) can be any real number between 0 and 2, inclusive.
03

Determine the Behavior of the Function in the Domain

Since \(f(x) = x^2\) is increasing in the interval \([0, 2]\), as \(x\) increases from 0 to 2, the value of \(f(x)\) also increases, beginning at 0 when \(x = 0\) and reaching 4 when \(x = 2\).
04

Find the Range

To find the range, evaluate \(f(x)\) at the boundaries of the domain. \(f(0) = 0\) and \(f(2) = 4\). Since the function is monotonically increasing in the given interval, the range of \(f(x)\) is all values from the minimum \(f(0)\) to the maximum \(f(2)\), hence the range is \([0, 4]\).
05

Graph the Function

Plot the points for \(x = 0\), \(x = 1\), and \(x = 2\) on the Cartesian plane. At \(x = 0\), \(f(x) = 0\); at \(x = 1\), \(f(x) = 1\); at \(x = 2\), \(f(x) = 4\). Since the points form part of a parabola, sketch the curve smoothly through these points, starting at point \((0, 0)\) and ending at point \((2, 4)\).

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Key Concepts

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

Quadratic Functions
Quadratic functions are a specific type of polynomial function where the highest degree of the variable is squared, represented as \( f(x) = ax^2 + bx + c \). In these functions, the graph forms a distinctive curve known as a parabola. A key feature of parabolas is their symmetry, centered around a vertical line called the axis of symmetry. This axis runs through the vertex of the parabola, which is the point where it turns. Quadratics have many important properties:
  • The direction the parabola opens (up or down) depends on the coefficient of the \( x^2 \) term (if \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards).
  • Vertex: This is either the highest or lowest point of the parabola, depending on the direction it opens.
  • The solutions to the equation \( f(x) = 0 \) (where the parabola crosses the x-axis) are known as roots or zeroes.
Graphing Functions
Graphing functions involves plotting points derived from input-output pairs and then connecting them to illustrate the overall shape of the function. For quadratic functions like \( f(x) = x^2 \), the process involves following the equation over its domain. To graph a function:
  • Start by selecting points within the domain. Here, the domain is the interval \([0, 2]\).
  • Calculate the corresponding \( f(x) \) values. For instance, at \( x = 0 \), \( f(x) = 0 \), and at \( x = 2 \), \( f(x) = 4 \).
  • Plot these points on a coordinate plane.
  • Since quadratic functions graph as parabolas, smoothly connect the plotted points.
Remember to consider the pattern: increasing functions rise as x increases, which is evident as \( f(x) \) rises from 0 to 4 over the domain.
Domain and Range
Understanding the domain and range is crucial when studying functions. The domain of a function is the complete set of possible input values (\( x \) values) that the function can accept without causing errors like division by zero or negative square roots. In our example, the domain is \([0, 2]\). This means only \( x \) values from 0 to 2 are considered.Range, in contrast, refers to the set of possible output values (\( f(x) \)) that the function can produce. For the quadratic \( f(x) = x^2 \) within this domain:
  • The smallest output (or minimum value) occurs at \( x = 0 \, (f(x) = 0)\).
  • The largest output is at \( x = 2 \, (f(x) = 4)\).
  • The function is continuous, and because \( f(x) \) consistently increases, the range includes all values falling between the minimum and maximum. Therefore, the range is \([0, 4]\).
Understanding both concepts helps in properly defining and working with functions.

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Most popular questions from this chapter

Bite Strength The bite strength of finches increases as body mass increases. Van de May and Bout (2004) made the following measurements of bird mass \((B\), measured in \(\mathrm{g}\) ) and wind beat frequency \((f\), measured in \(\mathrm{Hz})\) : \begin{tabular}{lcc} \hline Species & Body Mass \((\boldsymbol{B}\), Measured in g) & Bite Strength \((\boldsymbol{S}, \mathbf{M e a s u r e d ~ i n ~} \mathbf{N})\) \\ \hline Java sparrow & \(30.4\) & \(9.6\) \\ Red-billed firefinch & \(6.9\) & \(1.2\) \\ Double barred finch & \(9.7\) & \(1.9\) \\ \hline \end{tabular} Assume that there is a power-law dependence of \(S\) upon \(B\) : \(S=a B^{c}\), where \(a\) and \(c\) are some constants. By plotting \(\log S\) against log \(B\), estimate the parameters \(a\) and \(c\).

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