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Chapter 1: Problem 59

Find a simple function that fits the data in the tables. $$\begin{array}{|r|r|}\hline x & y \\\\\hline 0 & -1 \\\\\hline 1 & 0 \\\\\hline 4 & 1 \\\\\hline 9 & 2 \\\\\hline 16 & 3 \\ \hline\end{array}$$

Short Answer

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Question: Based on the given table of values, find the simple function that fits the data. Table: $$\begin{array}{|r|r|} \hline x & y \\\ \hline 0 & -1 \\\ \hline 1 & 0 \\ \hline 4 & 1 \\ \hline 9 & 2 \\ \hline 16 & 3 \\ \hline \end{array}$$

Step by step solution

01

Observe the relationship between x and y values

First, we should examine the given table and try to find a pattern or relationship between the x and y values. The table is: $$\begin{array}{|r|r|}\hline x & y \\\\\hline 0 & -1 \\\\\hline 1 & 0 \\\\\hline 4 & 1 \\\\\hline 9 & 2 \\\\\hline 16 & 3 \\\ \hline\end{array}$$ 鈥� When \(x=0\), \(y=-1\). 鈥� When \(x=1\), \(y=0\). 鈥� When \(x=4\), \(y=1\). 鈥� When \(x=9\), \(y=2\). 鈥� When \(x=16\), \(y=3\).
02

Identify the pattern

Notice that the x values are perfect squares and the y values sequentially increase by 1. Let's write the x values as perfect squares: 鈥� \(x=0^2\) when \(y=-1\). 鈥� \(x=1^2\) when \(y=0\). 鈥� \(x=2^2\) when \(y=1\). 鈥� \(x=3^2\) when \(y=2\). 鈥� \(x=4^2\) when \(y=3\).
03

Express the pattern as a function

We can see that the y value is one less than the square root of the x value. So, we can write the relationship between x and y as a function: $$y= \sqrt{x} -1 $$ Now, we have a simple function that fits the data from the table: $$f(x) = \sqrt{x} -1$$

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Key Concepts

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

Understanding Data Patterns
To find a function that fits data, we first need to spot a data pattern. Data patterns help us understand relationships between different data points. By identifying these patterns, we can form equations that express these relationships mathematically.
In this problem, it was crucial to analyze how the x values and y values change together and separately.
  • The x values in the table are 0, 1, 4, 9, 16. These numbers are not just random; they are perfect squares: 0虏, 1虏, 2虏, 3虏, 4虏.
  • The y values (-1, 0, 1, 2, 3) increment by 1 as x changes.
By writing x as square numbers, we reveal a clearer connection. This allows us to draw insights into how the pattern evolves, making function fitting more intuitive.
The Role of Mathematical Modeling
Mathematical modeling is a method to describe real-world problems using mathematical language and symbols. It involves translating observed data into a function or an equation. This model can then help predict or understand other similar situations.
In our case, after examining the data, we found a pattern involving perfect squares. We realized that the function expression that best describes the y values in relation to x is:\[ y = \sqrt{x} - 1 \]This model tells us that y is the square root of x, minus 1. When modeling mathematically, it's vital to ensure that our model aligns well with all given data points. This reliable transformation from data to model exemplifies how mathematical modeling provides clarity and forecasts.
The Significance of Square Roots in Function Fitting
Square roots are essential in the world of mathematics and play a crucial role in function fitting. A square root function takes a number as input and produces a value which, when squared, returns the original number. This operation is vital in patterns involving quadratic relationships.
For our table, identifying that the x values are squares allowed us to apply the square root operation effectively. By calculating the square root of x values, we were able to find a simple pattern:
  • Square root acts as the inverse of squaring, unraveling x and making the relationship with y evident.
  • The y value of each corresponding x value follows the formula \( y = \sqrt{x} - 1 \).
Grasping how square roots fit into this equation allowed us to develop a precise function, simplifying the relationship between x and y and demonstrating the power of basic mathematical concepts like square roots in uncovering data patterns.

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Most popular questions from this chapter

a. Find the linear function \(C=f(F)\) that gives the reading on the Celsius temperature scale corresponding to a reading on the Fahrenheit scale. Use the facts that \(C=0\) when \(F=32\) (freezing point) and \(C=100\) when \(F=212\) (boiling point). b. At what temperature are the Celsius and Fahrenheit readings equal?

Consider the quartic polynomial \(y=f(x)=x^{4}-x^{2}\). a. Graph \(f\) and estimate the largest intervals on which it is one-to-one. The goal is to find the inverse function on each of these intervals. b. Make the substitution \(u=x^{2}\) to solve the equation \(y=f(x)\) for \(x\) in terms of \(y .\) Be sure you have included all possible solutions. c. Write each inverse function in the form \(y=f^{-1}(x)\) for each of the intervals found in part (a).

A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function \(Q(t)=a\left(1-e^{-t / c}\right),\) where \(t\) is measured in seconds, and \(a\) and \(c > 0\) are physical constants. The steady-state charge is the value that \(Q(t)\) approaches as \(t\) becomes large. a. Graph the charge function for \(t \geq 0\) using \(a=1\) and \(c=10\) Find a graphing window that shows the full range of the function. b. Vary the value of \(a\) holding \(c\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(a ?\) c. Vary the value of \(c\) holding \(a\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(c ?\) d. Find a formula that gives the steady-state charge in terms of \(a\) and \(c\).

Modify Exercise 84 and use property \(\mathrm{E} 2\) for exponents to prove that \(\log _{b}(x / y)=\log _{b} x-\log _{b} y\).

Determine whether the following statements are true and give an explanation or counterexample. a. The range of \(f(x)=2 x-38\) is all real numbers. b. The relation \(f(x)=x^{6}+1\) is not a function because \(f(1)=f(-1)=2\) c. If \(f(x)=x^{-1},\) then \(f(1 / x)=1 / f(x)\) d. In general, \(f(f(x))=(f(x))^{2}\) e. In general, \(f(g(x))=g(f(x))\) f. In general, \(f(g(x))=(f \circ g)(x)\) g. If \(f(x)\) is an even function, then \(c f(a x)\) is an even function, where \(a\) and \(c\) are real numbers. h. If \(f(x)\) is an odd function, then \(f(x)+d\) is an odd function, where \(d\) is a real number. i. If \(f\) is both even and odd, then \(f(x)=0\) for all \(x\)
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