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Chapter 4: Problem 29

The table in the following figure shows the population of Los Angeles over the years \(1930-1990 .\) The accompanying graph shows the corresponding scatter plot and regression line. The equation of the regression line is $$ f(x)=37,546.068 x-71,238,863.429 $$(a) Use the regression line to compute an estimate for what the population of Los Angeles might have been in 2000 . (Round the answer to the nearest thousand.) Then compute the percentage error in the estimate, given that the actual figure for 2000 is 3.823 million. (b) Find \(f^{-1}(x)\) (c) Use your answer in part (b) to estimate the year in which the population of Los Angeles might reach 4 million. Hint: For the function \(f\), the inputs are years and the outputs are populations; for \(f^{-1}\), the inputs are populations and the outputs are years.

Short Answer

Expert verified
(a) Estimated population in 2000: 3,853,000; Percentage error: 0.785%. (b) \(f^{-1}(x) = \frac{x + 71,238,863.429}{37,546.068}\). (c) Estimated year for 4 million: 2004.

Step by step solution

01

Set up the equation for estimation

To estimate the population in the year 2000 using the regression line \(f(x) = 37,546.068x - 71,238,863.429\), we substitute \(x = 2000\) into the equation, because \(x\) represents the year.
02

Calculate the estimated population for 2000

Substitute \(x = 2000\) into the equation: \[ f(2000) = 37,546.068 \times 2000 - 71,238,863.429 \]Simplify the equation:\[ f(2000) = 75,092,136 - 71,238,863.429 = 3,853,272.571 \]Round to the nearest thousand: 3,853,000.
03

Calculate percentage error

First, find the difference between the estimated and actual population for 2000, which is 3.823 million (3,823,000):\[ \text{Error} = 3,853,000 - 3,823,000 = 30,000 \]Then, find the percentage error:\[ \text{Percentage Error} = \left(\frac{30,000}{3,823,000}\right) \times 100 \approx 0.785\% \]
04

Find the inverse function \(f^{-1}(x)\)

Start from the original function: \[ y = 37,546.068x - 71,238,863.429 \] To find \(f^{-1}(x)\), solve for \(x\): \[ y + 71,238,863.429 = 37,546.068x \] \[ x = \frac{y + 71,238,863.429}{37,546.068} \] Thus, \( f^{-1}(x) = \frac{x + 71,238,863.429}{37,546.068} \)
05

Estimate the year population might reach 4 million using \(f^{-1}(x)\)

Substitute \(x = 4,000,000\) into the inverse function found:\[ f^{-1}(4,000,000) = \frac{4,000,000 + 71,238,863.429}{37,546.068} \]Calculate:\[ f^{-1}(4,000,000) = \frac{75,238,863.429}{37,546.068} \approx 2004 \] Therefore, the population might reach 4 million around the year 2004.

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Key Concepts

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

Inverse Functions
Inverse functions are like a mirror image in mathematics. They can reverse the effect of the original function. In simple terms, if a function maps a value from a set to a new value, the inverse function takes that new value and maps it back to the original value.

Think of it like switching the input and output. If the original function represents a process or transformation, the inverse function reverses it. In our example, if you have a function that takes a year and outputs the population, the inverse function will take the population and output the year.

This idea is crucial when predicting when a future event might occur, like estimating the year when a population reaches a certain size. To find an inverse function algebraically, we swap the dependent and independent variables in the equation and then solve for the independent variable. This process allows us to calculate inputs based on given outputs, effectively reversing the original function's operations.
Percentage Error
Percentage error is a useful statistic that helps quantify how accurate an estimate is compared to an actual value. It offers insight into the reliability of a prediction or calculation.

To calculate percentage error, you subtract the actual value from the estimated value. This difference is the 'error'. Then, you divide the error by the actual value to convert it into a percentage, which makes it easier to grasp how big or small the error is compared to the actual value. Finally, multiply the result by 100 to express it as a percentage.
  • The formula is: \(\text{Percentage Error} = \left( \frac{\text{Estimated Value} - \text{Actual Value}}{\text{Actual Value}} \right) \times 100 \).
  • In practice, the percentage error provides a clear picture of the extent of overestimation or underestimation, offering a precise comparison of forecasted versus actual results.
Understanding percentage error is essential in many fields such as science, finance, and marketing, where precise estimates are crucial for making informed decisions.
Population Estimation
Population estimation is a technique used to predict the number of individuals in a given area at a certain time. It's especially useful for urban planning, resource allocation, and public policy. To make these predictions, analysts often use mathematical models like regression analysis.

Regression analysis lets us draw a trend line through historical data points to foresee future values. This trend line is represented by a mathematical equation known as the regression line. It captures the relationship between time and population growth efficiently. The equation of the regression line embodies this predictive model, where '*x*' is the year, and '*f(x)*' is the estimated population for that year.
  • In our example, the equation is \(f(x) = 37,546.068x - 71,238,863.429\).
  • By inputting any year into this equation, you can estimate what the population might be.
Population estimation through regression helps planners and policymakers make strategic decisions with potential for future development. Understanding the nuances of this technique can greatly aid in managing future changes effectively.

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Most popular questions from this chapter

Find quadratic functions satisfying the given conditions. The vertex is \((3,-1),\) and one \(x\) -intercept is 1.

Determine whether the \(x\) -y values are generated by a linear function, a quadratic function, or neither. $$\begin{array}{lccccc} \hline x & -4 & -2 & 0 & 2 & 4 \\ y & 25 & 3 & -4 & 7 & 33 \\ \hline \end{array}$$

The functions \(f, g,\) and h are defined as follows: $$ f(x)=2 x-3 \quad g(x)=x^{2}+4 x+1 \quad h(x)=1-2 x^{2} $$ In each exercise, classify the function as linear, quadratic, or neither. $$g \circ h$$

Let \(f(x)=\left(x^{5}+1\right) / x^{2}\) (a) Graph the function \(f\) using a viewing rectangle that extends from -4 to 4 in the \(x\) -direction and from -8 to 8 in the \(y\) -direction. (b) Add the graph of the curve \(y=x^{3}\) to your picture in part (a). Note that as \(|x|\) increases (that is, as \(x\) moves away from the origin), the graph of \(f\) looks more and more like the curve \(y=x^{3} .\) For additional perspective, first change the viewing rectangle so that \(y\) extends from -20 to \(20 .\) (Retain the \(x\) -settings for the moment.) Describe what you see. Next, adjust the viewing rectangle so that \(x\) extends from -10 to 10 and \(y\) extends from -100 to \(100 .\) Summarize your observations. (c) In the text we said that a line is an asymptote for a curve if the distance between the line and the curve approaches zero as we move further and further out along the curve. The work in part (b) illustrates that a curve can behave like an asymptote for another curve. In particular, part (b) illustrates that the distance between the curve \(y=x^{3}\) and the graph of the given function \(f\) approaches zero as we move further and further out along the graph of \(f .\) That is, the curve \(y=x^{3}\) is an "asymptote" for the graph of the given function \(f\). Complete the following two tables for a numerical perspective on this. In the tables, \(d\) denotes the vertical distance between the curve \(y=x^{3}\) and the graph of \(f:\) $$ d=\left|\frac{x^{5}+1}{x^{2}}-x^{3}\right| $$ $$\begin{array}{llllll} \hline x & 5 & 10 & 50 & 100 & 500 \\ \hline d & & & & \\ \hline & & & & \\ \hline x & -5 & -10 & -50 & -100 & -500 \\ \hline d & & & & \\ \hline \end{array}$$ (d) Parts (b) and (c) have provided both a graphical and a numerical perspective. For an algebraic perspective that ties together the previous results, verify the following identity, and then use it to explain why the results in parts (b) and (c) were inevitable: $$ \frac{x^{5}+1}{x^{2}}=x^{3}+\frac{1}{x^{2}} $$

(a) Determine the \(x\) - and \(y\) -intercepts and the excluded regions for the graph of the given function. Specify your results using a sketch similar to Figure \(16(a) .\) In Exercises \(31-34\) you will first need to factor the polynomial. (b) Graph each function. $$y=x^{3}-9 x$$
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