- The function \(H(x)\) defines the percentage of households with Internet as a function of the number of years (\(x\)) after 1980.
- \(H(27) = 61.7\) means that in 2007, 61.7% of households had Internet.
- \(H(4) = k\) represents the unknown percentage of households with Internet in 1984.
- \(H(37) > H(30)\) indicates that Internet usage was higher in 2017 than in 2010.
- \(H(17) + H(21) \approx H(29)\) shows that the sum of Internet usage percentages in 1997 and 2001 is approximately equal to the percentage in 2009.
Explanation
- Understanding the Function Notation
The function \(H(x)\) provides a way to describe the percentage of U.S. households with Internet use as a function of time. Here, \(x\) represents the number of years after 1980, and \(H(x)\) is the output representing the percentage. This means that to determine the specific calendar year for any given \(x\), we calculate the year as \(1980 + x\). Understanding this relationship allows us to interpret demographic and technological trends expressed through mathematical notation.
- Interpreting H(27) = 61.7
The expression \(H(27) = 61.7\) indicates that at 27 years after 1980, the percentage of U.S. households with Internet use was 61.7%. Calculating the year, we have \(1980 + 27 = 2007\). Thus, in 2007, 61.7% of U.S. households had Internet access.
- Interpreting H(4) = k
The expression \(H(4) = k\) refers to the situation 4 years after 1980, which corresponds to the year \(1980 + 4 = 1984\). Here, \(k\) is a variable representing the unknown percentage of U.S. households that had Internet access in 1984. Without further data, \(k\) remains the placeholder for that specific year's statistic.
- Interpreting H(37) > H(30)
The inequality \(H(37) > H(30)\) compares the percentages of Internet use in two different years. The value \(H(37)\) corresponds to the year \(1980 + 37 = 2017\), and the value \(H(30)\) corresponds to the year \(1980 + 30 = 2010\). The inequality states that the percentage of households with Internet access was higher in 2017 than it was in 2010.
- Interpreting H(17) + H(21) ≈ H(29)
The expression \(H(17) + H(21) \approx H(29)\) relates statistics from three different years. The value \(H(17)\) refers to the year \(1980 + 17 = 1997\), \(H(21)\) refers to the year \(1980 + 21 = 2001\), and \(H(29)\) refers to the year \(1980 + 29 = 2009\). The equation states that the sum of the percentage of households with Internet in 1997 and 2001 is approximately equal to the percentage of households with Internet in 2009.
Examples
Function notation is frequently used in fields like economics, public health, and urban planning to track trends over time. For example, if a city tracks the average commute time \(T(y)\) where \(y\) is the number of years after 2020, saying \(T(2) = 35\) would mean that two years after 2020 (in 2022), the average commute time was 35 minutes. Understanding this notation helps citizens and policy-makers quickly grasp reports on population growth, traffic density, or environmental changes, allowing for data-driven decisions.