Tamalpais Union High School District / Overview
Composition of Functions – Word Problems
Part 2: Simple Application
In the mail, you receive a coupon for $5 off of a pair of jeans. When you arrive at the store, you find that all jeans are 25% off.
Let x represent the original cost of the jeans.
1. Write a function, f(x), that represents the effect of your original coupon.
2. Write a function, g(x), that represents the effect of the 25% discount at the store.
3. Write a function, h(x), that represents how much you would pay if you use the mail coupon first followed by applying the discount from the store.
4. Write a function, j(x), that represents how much you would pay if you use the store discount first, followed by the mail coupon.
5. You find a pair of jeans for $36. How much would you pay for it using both functions h(x) and J(x).
6. If you only have $40 with you, what’s the most expensive pair of jeans you can purchase? (do not consider tax).
7. Determine when you would want to use h(x) applying the $5 coupon first, and determine when you would want to use j(x) applying the 25% off first.
Part 2: Composite Functions and their domain and range
Carrie, marine biologist is performing experiments along the continental slope off of the coast of baja California where the biodiversity is very dynamic. Scientists have proven that biodiversity is closely linked to the temperature of the water. Carrie wants to monitor the temperature of the ocean at different depths along the continental slope, to help record the changes in biodiversity due to changes in the temperature of the water. At the same time she does not want the robot to crash into the continental slope, so she needs to take into effects the speed of the currents.
Earlier scientists have found that the speed of ocean current as a function of depth. The speed, S, depends on depth, d, according to the following formula.
S(d) = 3d +1
Where S is measured in meters per second and d is measured in meters.
Suppose that the depth of a research robot depends on time, t, according to the formula:
d(t)=(1/27)t2
1. Use function composition to write the speed of the current at the depth of the robot as a function of time. Give an exact expression.
2. What is the speed of the current at the depth of the robot after 9 seconds? Round your answer, if necessary, to the nearest integer.
3. What is the realistic domain of the robot and what does that represent?
4. What is the realistic range of the robot and what does that represent? (It might help to find the vertex of d(t))
Day 12 Notes Composite Functions and Application
Objective: SWBAT find the composite of two functions and apply them to real-world problems
Function Review:
f(x) = 3x + 2
x = input = 4
f(4) =
Composite Functions -> the embedding of functions
f (x) = 3x + 2 g(x) = 4x – 1
f(g(x))
x = input = 3
f(g(3))
g(3) = 4(3) – 1 f(x) = 3x + 2
= 12 – 1 = 11 f(11)=
f(g(x)) = f(4x - 1) b/c g(x) = 4x – 1
= 3(4x – 1) + 2
= 12x – 3 + 2
= 12x – 1
Example 2: f (x) = x2 - 3 g(x) = 2x – 1
f(g(2)) = f(3) b/c g(2) = 2(2) -1 = 3
= 32 – 3
= 6
f(g(x)) = f(2x -1)
= (2x – 1)2 – 3
Part 1: Basic Practice
I. Let f(x) = 2x – 1, g(x) = 3x, and h(x) = x2 + 1. Compute the following:
1. f(g(x)) 2. f(h(x)) 3. g(h(x))
4. f(g(-3)) 5. f(h(7)) 6. g(h(0))
II. Let f(x) = 9 – x , g(x) = x2 + x, and h(x) = x – 2. Compute the following:
7. g(f(x)) 8. f(g(x)) 9. h(f(x))
10. g(f(-3)) 11. f(g(11)) 12. h(f(-6))
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A way of thinking about composite functions is taking the output of one function and making it the input of another. (Like a Double Function Machine)
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