Verified Answer :
- \(H(27) = 61.7\) translates to 61.7% of U.S. households having Internet use in 2007.
- \(H(4) = k\) translates to \(k\%\) of U.S. households having Internet use in 1984.
- \(H(37) > H(30)\) means Internet usage was higher in 2017 than in 2010.
- \(H(17) + H(21) \approx H(29)\) indicates the sum of usage in 1997 and 2001 is approximately the usage in 2009.
Explanation
- Interpreting Function Notation for U.S. Internet Usage
The function \(H(x)\) is defined as the percentage of U.S. households with Internet use \(x\) years after 1980. To interpret these statements, we must convert the input variable \(x\) into calendar years by calculating the year as \(1980 + x\).
- \(H(27) = 61.7\): Substituting \(x=27\), the year is \(1980 + 27 = 2007\). This means 61.7% of U.S. households had Internet use in 2007.
- \(H(4) = k\): Substituting \(x=4\), the year is \(1980 + 4 = 1984\). This means \(k\%\) of U.S. households had Internet use in 1984.
- \(H(37) > H(30)\): For \(x=37\), the year is \(2017\). For \(x=30\), the year is \(2010\). This statement indicates that the percentage of U.S. households with Internet use was greater in 2017 than it was in 2010.
- \(H(17) + H(21) \approx H(29)\): For \(x=17, 21,\) and \(29\), the corresponding years are \(1997, 2001,\) and \(2009\). This indicates that the sum of the percentages of U.S. households with Internet use in 1997 and 2001 is approximately equal to the percentage of households with Internet use in 2009.
- Matching Descriptive Statements
Now, we match the interpreted statements with the provided descriptive statements:
- \(H(27) = 61.7\) matches: "61.7% of U.S. households had Internet use in 2007."
- \(H(4) = k\) matches: "\(k\%\) of U.S. households had internet use in 1984."
- \(H(37) > H(30)\) matches: "The percent of U.S. households with Internet use was greater in 2017 than 2010."
- \(H(17) + H(21) \approx H(29)\) matches the description: "The sum of the percentages of U.S. households with Internet use in 1997 and 2001 is approximately equal to the percentage in 2009."
Examples
Function notation is essential for tracking trends over time, such as in economics, climate science, or social studies. For example, if a researcher defines \(C(t)\) as the cost of a gallon of gasoline \(t\) years after 2020, they can compare \(C(0)\) to \(C(3)\) to determine if fuel prices have risen or fallen over a three-year period. By translating these abstract functions into real-world dates and values, we can make informed decisions and observations about historical data and future predictions.