Math 2b/3a
Pfrommer
Name _________________________
9.16.14
Composites of functions
Imagine you are working on a science experiment about extreme temperatures. We have three
different temperature units that are commonly used: Fahrenheit, Celsius, and Kelvin. In the
United States we always measure temperature in Fahrenheit, but in most of the rest of the world,
temperature is measured in Celsius. In science, Kelvin is the primary unit of measurement used
in the physical sciences. Since there are multiple units used for temperature, there are also
formulas to convert from one unit to another.
K(c) = c – 273.16
C(f) =
5
(f - 32)
9
is used to convert temperature in Celsius to Kelvin, c represents the
temperature in Celsius.
is used to convert temperature in Fahrenheit to Celsius, f represents the
temperature in Fahrenheit.
Convert the following temperatures to Kelvin.
1. 30º C
2. 45º F
3. 80º F
4. 67º F
In exercises 2-4 you were composing the given functions. This means you input a number into
one of the functions and then took that result and put it as the input of the second function.
For #2, you found C(45º) which equaled approximately 7.22 and then you found K(7.22)
equaled -265.94 Kelvin. We often write this as K(C(f)) or K C(f).
Are you tired of doing multiple computations yet? If so, there is a short-cut! Let’s find a general
formula for K(C(f)).
In order to do this we plug in f into our function C. Guess what, that is already how it is written.
Now, we simply need to take our result: C(f) and plug that whole thing in for the input into K(c).
5
When we do this we get: (f - 32) - 273.16. We can simplify this by distributing the five9
5
5
ninths. Our result is f -17.78 - 273.16 = f - 290.94 . Let’s make sure that our new formula
9
9
gives the same result as when we did the problem in 2 steps. Let’s find K(C(45)) using our new
5
formula. (45) - 290.94 = -265.94 . That is the same result!
9
Exercises:
4. If f(x) = 2x – 5 and g(x) = x2, find the following:
a. f(g(3))
d. f(g(x))
b. g(f(3))
c. f(f(-2))
e. g(f(x))
f. why is it not surprising that the answers to parts a and b are not equal. Hint: look
at parts d and e.
HW: this worksheet & pg. 286 # 1a-e ,3, 12a-d, 17a,b
Math 2b/3a
Pfrommer
Name _________________________
9.16.14
Exercises:
1. Find these composite formulas, using the method from part c of the example above.
a. Suppose f(x) = 5x – 3 and g(x) = 2x + 7. Write the formula for g(f(x)).
b. Suppose f(x) = x3 and g(x) =
x . Write the formula for g(f(x)).
2. For each pair of functions f(x) and g(x), find f(g(3)) and g(f(-2)).
a. f(x) = 4x – 7, g(x) = x2
b. f(x) = x3,
c. f(x) =
x,
g(x) = 2x + 6
g(x) = 1 + x
d. f(x) = 3x,
g(x) = 5x – 1
e. f(x) = 4x,
g(x) = x + x2
3. Consider the composite where the first function is f(x) = 3 – x and the second is g(x) = 2x.
a. Evaluate g(f(1)).
b.
Evaluate g(f(3)).
c. Complete this input-output table for the composite. You found two of the numbers
already in parts a and b.
x
1
2
3
4
5
g(f(x))
4. Use these functions for all problems on this page:
f(x) = x – 2, g(x) = x .
a. Evaluate g(f(6)).
b. Evaluate g(f(11)).
c. Fill in this input-output table for the composite. You will need to find some of the needed
square-roots on your calculator (OK to round to three decimal places).
x
2
4
6
8
10
g(f((x))
d. Try to evaluate g(f(1)). Something goes wrong. Explain.
e. The point of part d is that x = 1 cannot be in the domain for this composite function.
What x values can be included in the domain?
HW: this worksheet & pg. 286 # 1a-e ,3, 12a-d, 17a,b