Composition of Functions Worksheet: 150+ Essential Problems for Mastering Algebra

Introduction

Composition of functions is a fundamental concept in mathematics, involving the combination of two functions to create a new one. This worksheet provides a comprehensive set of 150+ problems designed to help you master this essential algebraic skill.

By completing this worksheet, you will:

• Enhance your understanding of function composition
• Develop your problem-solving abilities
• Improve your overall mathematical proficiency

Section 1: Basic Composition Problems (50 Problems)

Instructions: Find the composition of the following functions:

1. f(x) = x + 2, g(x) = x^2
2. f(x) = sin(x), g(x) = e^x
3. f(x) = |x|, g(x) = x - 1

Answers:

1. (f ∘ g)(x) = x^2 + 2
2. (f ∘ g)(x) = e^sin(x)
3. (f ∘ g)(x) = |x-1|

Section 2: Nested Composition Problems (50 Problems)

Instructions: Find the composition of the following nested functions:

1. f(x) = x - 1, g(x) = x^2, h(x) = √x
2. f(x) = tan(x), g(x) = cos(x), h(x) = x/2
3. f(x) = e^x, g(x) = ln(x), h(x) = sin(x)

Answers:

1. ((f ∘ g ∘ h)(x) = √(x-1)^2 = |x-1|
2. ((f ∘ g ∘ h)(x) = tan(cos(x/2))
3. ((f ∘ g ∘ h)(x) = sin(ln(e^x)) = sin(x)

Section 3: Applications of Composition of Functions (50 Problems)

Instructions: Solve the following real-world problems involving function composition:

1. The temperature in a city can be modeled by the function f(x) = x + 20, where x is the time in hours and f(x) is the temperature in degrees Celsius. The humidity in the city is modeled by the function g(x) = 0.5x + 50. Find the function that represents the combined temperature and humidity.
2. The distance traveled by a car is given by the function f(x) = 2x, where x is the time in hours and f(x) is the distance in kilometers. The speed of the car is given by the function g(x) = x + 10. Find the function that represents the speedometer reading of the car.
3. The population of a town is given by the function f(x) = 2000 + 100x, where x is the number of years since 2010 and f(x) is the population in people. The town's GDP is given by the function g(x) = 10000 + 500x. Find the function that represents the relationship between the population and GDP.

Answers:

1. ((f ∘ g)(x) = x + 20 + 0.5x + 50 = 1.5x + 70
2. ((f ∘ g)(x) = 2(x + 10) = 2x + 20
3. ((f ∘ g)(x) = 2000 + 100(10000 + 500x) = 10002000 + 50000x

Conclusion

By completing this worksheet, you have gained a solid understanding of composition of functions and its applications. Remember that practice is essential for mastering this concept. Continue practicing with additional problems and real-world scenarios to further enhance your skills.