Worksheet A Topic 2.7 Composition Of Functions

Worksheet: Topic 2.7 Composition of Functions

This comprehensive guide delves into the topic of composition of functions, a crucial concept in algebra and precalculus. We'll explore the definition, notation, methods for finding compositions, domain and range considerations, and finally, solidify our understanding with a variety of examples and practice problems. This worksheet is designed to be a complete resource, helping you master this essential mathematical skill.

Understanding Composition of Functions

Function composition is essentially the act of applying one function to the output of another function. Instead of simply evaluating a function at a single value, we're nesting one function inside another. Think of it like a pipeline where the output of one stage becomes the input of the next.

Notation and Terminology

The composition of two functions, f(x) and g(x), is denoted in two primary ways:

• (f ∘ g)(x): This reads as "f of g of x" or "f composed with g of x". This notation emphasizes that we apply g first, and then apply f to the result.

• f(g(x)): This notation, while less formal, clearly illustrates the order of operations. We evaluate g(x) first, and then substitute the result into f(x).

It's crucial to understand that (f ∘ g)(x) ≠ (g ∘ f)(x) in most cases. The order of composition significantly impacts the final result. Composition of functions is not commutative.

Methods for Finding Compositions

Let's explore the process of finding compositions with several examples.

Example 1: Simple Composition

Let f(x) = x² + 1 and g(x) = 2x. Find (f ∘ g)(x) and (g ∘ f)(x).

(f ∘ g)(x) = f(g(x)) = f(2x) = (2x)² + 1 = 4x² + 1

(g ∘ f)(x) = g(f(x)) = g(x² + 1) = 2(x² + 1) = 2x² + 2

Notice how (f ∘ g)(x) and (g ∘ f)(x) produce different results.

Example 2: Composition with More Complex Functions

Let f(x) = √(x - 3) and g(x) = x² + 2. Find (f ∘ g)(x) and (g ∘ f)(x). Pay close attention to the domain restrictions!

(f ∘ g)(x) = f(g(x)) = f(x² + 2) = √((x² + 2) - 3) = √(x² - 1)

The domain of (f ∘ g)(x) is restricted because the expression inside the square root must be non-negative: x² - 1 ≥ 0, implying x ≤ -1 or x ≥ 1.

(g ∘ f)(x) = g(f(x)) = g(√(x - 3)) = (√(x - 3))² + 2 = x - 3 + 2 = x - 1

The domain of (g ∘ f)(x) is limited by the domain of f(x): x - 3 ≥ 0, meaning x ≥ 3.

Domain and Range of Composite Functions

Determining the domain and range of composite functions requires careful consideration. The domain of (f ∘ g)(x) is restricted not only by the domain of f(x) but also by the range of g(x). Any value in the range of g(x) that is not in the domain of f(x) must be excluded from the domain of (f ∘ g)(x).

Example 3: Domain Restrictions in Composition

Let f(x) = 1/x and g(x) = x - 2. Find the domain of (f ∘ g)(x).

(f ∘ g)(x) = f(g(x)) = f(x - 2) = 1/(x - 2)

The domain of (f ∘ g)(x) is all real numbers except x = 2, because division by zero is undefined.

Composition of More Than Two Functions

The concept of composition can be extended to more than two functions. For instance, we can find (f ∘ g ∘ h)(x), which means applying h first, then g, and finally f.

Example 4: Composition of Three Functions

Let f(x) = x + 1, g(x) = 2x, and *h(x) = x². Find (f ∘ g ∘ h)(x).

(f ∘ g ∘ h)(x) = f(g(h(x))) = f(g(x²)) = f(2x²) = 2x² + 1

Applications of Composition of Functions

Composition of functions is not merely a theoretical exercise; it has significant applications in various fields:

• Modeling Real-World Phenomena: Composition allows us to create more complex models by combining simpler functions. For instance, we could model the population growth of a species considering factors like resource availability and predation using a combination of exponential and logistic functions.

• Transformations in Geometry: Transformations such as translations, rotations, and scaling can be represented as functions. Composition of these functions describes the combined effect of multiple transformations.

• Computer Programming: Functions are fundamental building blocks in programming. Composition allows programmers to create more complex functions from simpler, reusable ones, promoting modularity and efficiency.

Practice Problems

Here are some practice problems to test your understanding:

1. Given f(x) = 3x + 2 and g(x) = x², find:

   • (f ∘ g)(x)
   • (g ∘ f)(x)
   • (f ∘ f)(x)
   • (g ∘ g)(x)

2. Given f(x) = √x and g(x) = x - 4, find:

   • (f ∘ g)(x) and its domain.
   • (g ∘ f)(x) and its domain.

3. Given f(x) = 1/(x+1) and g(x) = 2x - 1, find:

   • (f ∘ g)(x) and its domain.
   • (g ∘ f)(x) and its domain.

4. Given f(x) = x³, g(x) = x + 2, and h(x) = √x, find (f ∘ g ∘ h)(x) and its domain.

5. Let f(x) = x² and g(x) = x + 1. If (f ∘ g)(x) = 16, find the possible values of x.

Solutions to Practice Problems

(Solutions are provided below. Try to solve them independently before checking!)

1. • (f ∘ g)(x) = 3x² + 2
   • (g ∘ f)(x) = (3x + 2)² = 9x² + 12x + 4
   • (f ∘ f)(x) = 3(3x + 2) + 2 = 9x + 8
   • (g ∘ g)(x) = (x²)² = x⁴
2. • (f ∘ g)(x) = √(x - 4); Domain: x ≥ 4
   • (g ∘ f)(x) = √x - 4; Domain: x ≥ 0
3. • (f ∘ g)(x) = 1/(2x); Domain: x ≠ 0
   • (g ∘ f)(x) = 2/(x + 1) - 1 = (1 - x)/(x + 1); Domain: x ≠ -1

4. (f ∘ g ∘ h)(x) = (√x + 2)³; Domain: x ≥ 0

5. (f ∘ g)(x) = (x + 1)² = 16. This gives x + 1 = ±4, so x = 3 or x = -5.

This comprehensive worksheet provides a thorough understanding of composition of functions, including notation, methods, domain/range considerations, and practical applications. Remember to practice regularly to master this important mathematical concept. Good luck!