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Chapter 1: Problem 85

Match the data with one of the following functions $$f(x)=c x, g(x)=c x^{2}, h(x)=c \sqrt{|x|}, \quad \text {and} \quad r(x)=\frac{c}{x}$$ and determine the value of the constant \(c\) that will make the function fit the data in the table. $$\begin{array}{|c|c|c|c|c|c|}\hline x & -4 & -1 & 0 & 1 & 4 \\\\\hline y & -32 & -2 & 0 & -2 & -32 \\\\\hline \end{array}$$

Short Answer

Expert verified
The function that fits the data from the table is \(g(x) = c x^{2}\) with \(c = -2\).

Step by step solution

01

Identify the Function

Analyze the symmetry of the data in the table around \(x = 0\) which indicates that the function is even. From the given functions, \(g(x)\) and \(r(x)\) are the only even functions which can produce symmetrical y-values around \(x = 0\). The function \(r(x)\) can be eliminated because it would result in undefined value as \(x\) gets closer to zero whereas in our data, y equals zero when \(x\) equals zero. Therefore, the data matches with the function \(g(x)\) which is \(g(x) = c x^{2}\).
02

Determine the Constant \(c\)

After successfully identifying the function fitting the data, the constant \(c\) needs to be determined. This can be done by substituting one pair of \((x, y)\) values from the table into the function. When \(x = -4\), \(y = -32\), insert these into function \(g(x) = c x^{2}\) and solve for \(c\). Solving this gives \(c = -2\). So, the final function fitting the data will be \(g(x) = -2 x^{2}\).

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Key Concepts

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

Even and Odd Functions
The concept of even and odd functions is fundamental in understanding the symmetry of a function in relation to the y-axis. An even function is symmetric around the y-axis, which means that it meets the condition that for every number x in its domain, the number -x is also in the domain, and the function satisfies the equation
\( f(x) = f(-x) \).

Examples of even functions include \( x^2 \) and \( cos(x) \). Conversely, an odd function is such that it is symmetric with respect to the origin, fulfilling the relationship
\( f(-x) = -f(x) \).

Functions like \( x^3 \) and \( sin(x) \) demonstrate odd symmetry. When faced with exercises involving fitting functions to data, recognizing these properties can be a powerful tool. For instance, in our problem, by noticing that the data showed symmetry around \( x = 0 \), we could quickly deduce that the function must be even. This allowed us to eliminate some of the potential functions in question and focus on the right candidates.
Identifying Functions from Tables
Determining the type of function from a set of data points in a table requires careful analysis of the input-output relationship. Often, this task involves looking for patterns in how the y-values change as x-values vary. To effectively identify the function that describes a set of data, it's crucial to:
  • Examine the x-values and their corresponding y-values for consistency.
  • Look for symmetry or repeating patterns that indicate certain function properties.
  • Consider the behavior of the function as x approaches zero or other significant values to rule out functions that do not fit the observed data.
In the exercise, we scrutinized the table and noticed that the y-values were symmetrical around the y-axis for \( x = 0 \). This observation helped us to single out the even functions from the list provided. It's essential to use these analytical strategies to correctly match the data with the suitable function.
Determining Constants in Functions
When a function is known, but it contains an undetermined constant, it's necessary to figure out this constant to fully describe the relationship between variables. The first step is identifying the correct form of the function from data, as previously discussed. Once that's accomplished, you solve for the unknown constant by plugging in a pair of x and y values that the function must adhere to.

For example, in our exercise, after identifying \( g(x) = c x^{2} \) as the candidate function based on the observed even symmetry, we then chose a pair of \( (x, y) \) from the table. By substituting \( x = -4 \) and \( y = -32 \) into the equation and solving for \( c \) we discovered that \( c = -2 \). The resulting function \( g(x) = -2 x^{2} \) is the particular solution to our problem, describing the data perfectly. This process of identifying and solving for constants is a frequent requirement when modelling real-world scenarios with functions, making it an invaluable skill in both mathematics and applied sciences.

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Most popular questions from this chapter

Consider \(f(x)=\sqrt{x-2}\) and \(g(x)=\sqrt[3]{x-2}\) Why are the domains of \(f\) and \(g\) different?

Determine whether the statement is true or false. Justify your answer. A piecewise-defined function will always have at least one \(x\) -intercept or at least one \(y\) -intercept.

The table shows the monthly revenue \(y\) (in thousands of dollars) of a landscaping business for each month of the year \(2013,\) with \(x=1\) representing January. $$\begin{array}{|c|c|}\hline \text { Month, \(x\) } & \text { Revenue, \(y\) } \\\\\hline 1 & 5.2 \\\2 & 5.6 \\\3 & 6.6 \\ 4 & 8.3 \\\5 & 11.5 \\\6 & 15.8 \\\7 & 12.8 \\\8 & 10.1 \\\9 & 8.6 \\\10 & 6.9 \\\11 & 4.5 \\\12 & 2.7 \\\\\hline \end{array}$$ A mathematical model that represents these data is \(f(x)=\left\\{\begin{array}{l}-1.97 x+26.3 \\ 0.505 x^{2}-1.47 x+6.3\end{array}\right.\) (a) Use a graphing utility to graph the model. What is the domain of each part of the piecewise-defined function? How can you tell? Explain your reasoning. (b) Find \(f(5)\) and \(f(11),\) and interpret your results in the context of the problem. (c) How do the values obtained from the model in part (a) compare with the actual data values?

(a) Write the linear function \(f\) such that it has the indicated function values and (b) Sketch the graph of the function. $$f(-5)=-1, \quad f(5)=-1$$

A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of the problem. Let \(h\) represent the height of the balloon and let \(d\) represent the distance between the balloon and the receiving station. (b) Write the height of the balloon as a function of \(d\) What is the domain of the function?
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