standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Task
The table below shows some input-output pairs of two functions $f$ and $g$ that agree for the values that are given but some of their output values are missing.
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$t$"}], "td": [{"_value": "0"}, {"_value": "15"}, {"_value": "30"}, {"_value": "45"}, {"_value": "60"}, {"_value": "75"}, {"_value": "90"}, {"_value": "105"}, {"_value": "120"}]}, {"th": [{"_value": "$f(t)$"}], "td": [{"_value": "0"}, {"_value": "0.5"}, {}, {"_value": "1.3"}, {"_value": "2"}, {"_value": "2.7"}, {}, {"_value": "4"}, {}]}, {"th": [{"_value": "$g(t)$"}], "td": [{"_value": "0"}, {"_value": "0.5"}, {}, {"_value": "1.3"}, {"_value": "2"}, {"_value": "2.7"}, {}, {"_value": "4"}, {}]}]}]}]}
Complete the table in a way so that $f$ could be invertible and so that $g$ is definitely not invertible.
Graph both functions and explain from the graph why $f$ is invertible and $g$ is not.
Come up with two real life situations that $f$ and $g$ could be representing.
Find and interpret the value $f^{-1}(4)$ in terms of these contexts.


aligns: no