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“Composition of Functions”
Concept
Real-world situations can often be modeled by composing functions. In this activity you will use your higher order thinking skills to do exactly this.
Example
In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second, then the volume is given by the function:
where t is in seconds and v is in cubic millimeters. But if we want to know the radius of the balloon, then we use the function:
where v is in cubic millimeters and r is in millimeters. Now if we look at the composite function:
we have a function that will give us the radius of the balloon after t seconds.
1. The formula for converting Celsius temperature (x) to Kelvin temperature is
and the formula for converting Fahrenheit (x) to Celsius is
Find a formula for converting Fahrenheit to Kelvin.
2. A high-altitude spherical weather balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.03 inches per second and that r=48 inches at time t=0. Determine a function that models the volume V of the balloon at time t and find the volume at t=300 seconds.
3. Matthew stores a small cache of 4-inch diameter snowballs in the basement freezer, unaware that the freezer’s self-defrosting feature will cause each snowball to lose about 1 cubic inch of volume every 40 days. Find a function that will yield the diameter of each snowball after x days and determine the diameter of the snowballs after 360 days.
“Composition of Functions ” Grading Rubric
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TRAITS |
1 - INADEQUATE |
2 - MARGINAL |
3 - Satisfactory |
4 - Proficient |
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Strategy/ Procedures |
Does not use any strategy, or uses the wrong strategy to solve the problem, resulting in an incorrect answer |
Uses correct strategy to solve the problem, but makes aslight math error |
Uses a correct strategy to solve the problem |
Uses a correct and efficient strategy to solve the problem |
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Mathematical Concepts |
Shows no understanding of the concepts needed to solve the problem, resulting in an incorrect answer |
Shows some understanding of the concepts needed to solve the problem, but makes aslight math error |
Shows good understanding of the concepts needed to solve the problem |
Shows complete understanding of the concepts needed to solve the problem |
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Mathematical Terminology and Symbols |
Mathematical and/or terminology symbols are used incorrectly |
Mathematical and/or terminology symbols are used sloppily, |
Correct mathematical and/or terminology symbols are used |
Advanced, correct mathematical and/or terminology symbols are used |
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making difficult |
it to see |
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what was done |
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Neatness and Organization |
The work is very sloppy and unorganized |
The work is fairly organized, but is difficult to read |
The work is neat and organized |
The work is neat, detailed, and organized |
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