Composition Of Functions Examples With Answers

Composition of Functions: Examples with Answers

Understanding composition of functions is crucial for mastering advanced mathematical concepts. This comprehensive guide will delve into the intricacies of function composition, providing numerous examples with detailed solutions to solidify your understanding. We'll explore various scenarios, from simple polynomial functions to more complex scenarios involving trigonometric and logarithmic functions. By the end, you’ll be confident in composing and evaluating composite functions.

What is Composition of Functions?

Function composition is a mathematical operation that combines two or more functions to create a new function. Instead of applying functions sequentially, composition applies one function's output as the input for another. This is represented symbolically as (f ∘ g)(x) or f(g(x)), read as "f composed with g of x". It means we first apply function g to x, and then use the result as the input for function f.

Key Concept: The domain of the composite function (f ∘ g)(x) is restricted by both the domain of g(x) and the domain of f(x). The range of g(x) must be a subset of the domain of f(x) for the composition to be defined.

Simple Examples of Function Composition

Let's start with some straightforward examples to grasp the fundamental idea.

Example 1:

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

Solution:

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

Notice that (f ∘ g)(x) ≠ (g ∘ f)(x). Function composition is generally not commutative. The order in which you compose functions significantly impacts the resulting function.

Example 2:

Let f(x) = 2x and g(x) = x - 3. Find (f ∘ g)(2) and (g ∘ f)(2).

Solution:

• (f ∘ g)(x) = f(g(x)) = f(x - 3) = 2(x - 3) = 2x - 6
• (f ∘ g)(2) = 2(2) - 6 = -2
• (g ∘ f)(x) = g(f(x)) = g(2x) = 2x - 3
• (g ∘ f)(2) = 2(2) - 3 = 1

Again, we see the non-commutative nature of function composition.

Composition with More Complex Functions

Now, let's progress to examples incorporating more intricate functions.

Example 3: Polynomial and Linear Functions

Let f(x) = x³ - 2x + 1 and g(x) = 3x - 1. Find (f ∘ g)(x).

Solution:

(f ∘ g)(x) = f(g(x)) = f(3x - 1) = (3x - 1)³ - 2(3x - 1) + 1. Expanding this expression:

(3x - 1)³ = 27x³ - 27x² + 9x - 1

Therefore, (f ∘ g)(x) = 27x³ - 27x² + 9x - 1 - 6x + 2 + 1 = 27x³ - 27x² + 3x + 2

Example 4: Involving Trigonometric Functions

Let f(x) = sin(x) and g(x) = x². Find (f ∘ g)(x) and (g ∘ f)(x).

Solution:

• (f ∘ g)(x) = f(g(x)) = f(x²) = sin(x²)
• (g ∘ f)(x) = g(f(x)) = g(sin(x)) = (sin(x))² = sin²(x)

Here, we see how composing trigonometric functions with other types of functions can lead to new trigonometric expressions.

Example 5: Composition with Logarithmic Functions

Let f(x) = ln(x) and g(x) = eˣ. Find (f ∘ g)(x) and (g ∘ f)(x). (Remember, ln(x) is the natural logarithm, base e.)

Solution:

• (f ∘ g)(x) = f(g(x)) = f(eˣ) = ln(eˣ) = x (The natural logarithm and exponential function are inverses of each other)
• (g ∘ f)(x) = g(f(x)) = g(ln(x)) = e^(ln(x)) = x (Again demonstrating the inverse relationship)

This example showcases the concept of inverse functions and how composition can reveal this relationship. Note that the domain for (g ∘ f)(x) is x > 0 because the natural logarithm is only defined for positive numbers.

Finding the Inverse and Composition

The concept of inverse functions is closely tied to function composition. If f(x) and g(x) are inverse functions, then (f ∘ g)(x) = x and (g ∘ f)(x) = x (for all x in the appropriate domains).

Example 6: Verifying Inverse Functions through Composition

Let f(x) = 2x + 3 and g(x) = (x - 3)/2. Verify if they are inverse functions.

Solution:

• (f ∘ g)(x) = f(g(x)) = f((x - 3)/2) = 2((x - 3)/2) + 3 = x - 3 + 3 = x
• (g ∘ f)(x) = g(f(x)) = g(2x + 3) = ((2x + 3) - 3)/2 = 2x/2 = x

Since both compositions result in x, f(x) and g(x) are indeed inverse functions.

Dealing with Domain Restrictions in Composition

It’s crucial to consider domain restrictions when composing functions. The domain of the composite function is determined by both the input function's domain and the output function's restrictions.

Example 7: Domain Restrictions

Let f(x) = √x and g(x) = x - 4. Find the domain of (f ∘ g)(x).

Solution:

(f ∘ g)(x) = f(g(x)) = f(x - 4) = √(x - 4)

The square root function is only defined for non-negative values. Therefore, x - 4 ≥ 0, which means x ≥ 4.

Thus, the domain of (f ∘ g)(x) is x ≥ 4.

Example 8: More Complex Domain Restrictions

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

Solution:

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

The denominator cannot be zero: 1/x - 2 ≠ 0 implies 1/x ≠ 2, so x ≠ 1/2. *Also, the denominator of the original function g(x) cannot be zero, so x ≠ 0. *Finally, the denominator of the composed function cannot be zero so 1-2x ≠ 0 which implies x ≠ 1/2

Therefore, the domain of (f ∘ g)(x) is all real numbers except x = 0 and x = 1/2.

Applications of Function Composition

Function composition finds applications in various fields:

• Physics: Modeling complex physical systems often involves combining simpler functions.
• Computer Science: Function composition is fundamental in programming and software design.
• Economics: Modeling economic behavior often relies on combining multiple functions.
• Calculus: Derivatives and integrals of composite functions are important concepts in calculus.

Conclusion

Mastering function composition is essential for success in higher-level mathematics and its various applications. By understanding the process of composing functions and carefully considering domain restrictions, you can confidently work with a wide range of functions and explore more advanced mathematical concepts. Remember to practice regularly with diverse examples to solidify your understanding and build your problem-solving skills. This detailed guide, with its numerous examples and solutions, should provide you with a strong foundation in the art of function composition.