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The graph below shows the percentage of people 25 years and older in the US who have completed a Bachelor's Degree over a several year period. Call this function f(t), where t is measured in years since 2006. Percent 29.5 29.0 28.5 28.0 27.5 27.0 26.5 2006 2007 - People 25 Years and Over Who Have Completed a Bachelor's Degree for the United States 2008 2009 2010 a. Explain how you know that this function has an inverse function. 2011 b. Give the domain and range of the inverse function. Find f-¹(27.5)and give a practical interpretation. 2012

The graph below shows the percentage of people 25 years and older in the US who have completed a Bachelor's Degree over a several year period. Call this function f(t), where t is measured in years since 2006. Percent 29.5 29.0 28.5 28.0 27.5 27.0 26.5 2006 2007 - People 25 Years and Over Who Have Completed a Bachelor's Degree for the United States 2008 2009 2010 a. Explain how you know that this function has an inverse function. 2011 b. Give the domain and range of the inverse function. Find f-¹(27.5)and give a practical interpretation. 2012

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Educational Website Transcription** **2. Graph and Function Analysis** The graph below illustrates the percentage of people aged 25 years and older in the United States who have completed a Bachelor's Degree over a multi-year period. This graph represents the function \( f(t) \), where \( t \) is measured in years since 2006. **Graph Details:** - **Title:** People 25 Years and Over Who Have Completed a Bachelor's Degree for the United States - **X-Axis:** Year (ranging from 2006 to 2012) - **Y-Axis:** Percentage of population (ranging from 26.5% to 29.5%) - **Line Trend:** Shows a steady increase from approximately 27.0% in 2006 to over 28.5% in 2012. **Analysis Questions:** **a. Explain how you know that this function has an inverse function.** To determine if a function has an inverse, it must be one-to-one (bijective), meaning each input maps to a unique output. Graphically, the function must pass the horizontal line test, where no horizontal line intersects the graph more than once. Observing the graph, the line is continuously increasing, which indicates that each year corresponds to a distinct percentage. Thus, this function is one-to-one and has an inverse. **b. Give the domain and range of the inverse function. Find \( f^{-1}(27.5) \) and give a practical interpretation.** - **Domain of the Inverse Function:** The range of the original function, which is approximately 27% to 29.5%. - **Range of the Inverse Function:** The domain of the original function, which is the years from 2006 to 2012. To find \( f^{-1}(27.5) \), locate the point on the graph where the percentage is 27.5%. This occurs around the year 2007. This means in 2007, the percentage of individuals aged 25 and older in the U.S. with a Bachelor's Degree was approximately 27.5%. **Practical Interpretation:** In the year 2007, 27.5% of the U.S. population aged 25 and older had completed a Bachelor's Degree.
Transcribed Image Text:**Educational Website Transcription** **2. Graph and Function Analysis** The graph below illustrates the percentage of people aged 25 years and older in the United States who have completed a Bachelor's Degree over a multi-year period. This graph represents the function \( f(t) \), where \( t \) is measured in years since 2006. **Graph Details:** - **Title:** People 25 Years and Over Who Have Completed a Bachelor's Degree for the United States - **X-Axis:** Year (ranging from 2006 to 2012) - **Y-Axis:** Percentage of population (ranging from 26.5% to 29.5%) - **Line Trend:** Shows a steady increase from approximately 27.0% in 2006 to over 28.5% in 2012. **Analysis Questions:** **a. Explain how you know that this function has an inverse function.** To determine if a function has an inverse, it must be one-to-one (bijective), meaning each input maps to a unique output. Graphically, the function must pass the horizontal line test, where no horizontal line intersects the graph more than once. Observing the graph, the line is continuously increasing, which indicates that each year corresponds to a distinct percentage. Thus, this function is one-to-one and has an inverse. **b. Give the domain and range of the inverse function. Find \( f^{-1}(27.5) \) and give a practical interpretation.** - **Domain of the Inverse Function:** The range of the original function, which is approximately 27% to 29.5%. - **Range of the Inverse Function:** The domain of the original function, which is the years from 2006 to 2012. To find \( f^{-1}(27.5) \), locate the point on the graph where the percentage is 27.5%. This occurs around the year 2007. This means in 2007, the percentage of individuals aged 25 and older in the U.S. with a Bachelor's Degree was approximately 27.5%. **Practical Interpretation:** In the year 2007, 27.5% of the U.S. population aged 25 and older had completed a Bachelor's Degree.
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