Composition And Inverses Of Functions Worksheet Answers

Composition and Inverses of Functions Worksheet Answers: A Comprehensive Guide

This comprehensive guide provides detailed explanations and solutions to common problems encountered in worksheets on composition and inverses of functions. We'll cover the fundamental concepts, walk through various examples, and equip you with the tools to tackle any related problem confidently. Understanding these concepts is crucial for success in algebra and higher-level mathematics.

Understanding Function Composition

Function composition is a fundamental operation in mathematics where the output of one function becomes the input of another. It's like a chain reaction, where the result of one step feeds into the next. We denote the composition of function f with function g as (f ∘ g)(x) or f(g(x)). This means we first apply function g to x, and then we apply function f to the result.

Key Points:

• Order Matters: Function composition is not commutative. (f ∘ g)(x) is generally not equal to (g ∘ f)(x).
• Domain Considerations: The domain of the composite function (f ∘ g)(x) is restricted by the domains of both f(x) and g(x). The range of g(x) must be within the domain of f(x).
• Notation: Pay close attention to notation; the order in which functions are composed significantly impacts the final result.

Example 1:

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

Solution:

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

Here, we substitute x² (the output of g(x)) into the function f(x).

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

In this case, we substitute (2x + 1) (the output of f(x)) into the function g(x). Notice that (f ∘ g)(x) ≠ (g ∘ f)(x).

Finding the Inverse of a Function

The inverse of a function, denoted as f⁻¹(x), "undoes" what the original function does. If f(a) = b, then f⁻¹(b) = a. Not all functions have inverses; a function must be one-to-one (or injective) to have an inverse. A one-to-one function means that each input maps to a unique output, and vice versa. You can visually check if a function is one-to-one using the horizontal line test: if any horizontal line intersects the graph more than once, the function is not one-to-one and doesn't have an inverse.

Steps to Find the Inverse:

1. Replace f(x) with y: This simplifies the notation.
2. Swap x and y: This is the crucial step that reverses the function's action.
3. Solve for y: Algebraically manipulate the equation to isolate y.
4. Replace y with f⁻¹(x): This denotes the inverse function.

Example 2:

Find the inverse of f(x) = 3x - 6.

Solution:

1. y = 3x - 6
2. x = 3y - 6 (Swapped x and y)
3. x + 6 = 3y
4. y = (x + 6)/3
5. f⁻¹(x) = (x + 6)/3

Therefore, the inverse of f(x) = 3x - 6 is f⁻¹(x) = (x + 6)/3.

Composition and Inverse Functions: A Powerful Combination

The relationship between a function and its inverse is particularly elegant when considering composition. If a function f(x) has an inverse f⁻¹(x), then:

• (f ∘ f⁻¹)(x) = x
• (f⁻¹ ∘ f)(x) = x

This means that composing a function with its inverse results in the identity function, which simply returns the input value. This property is a crucial check to verify if you have correctly found the inverse of a function.

Example 3:

Verify that f⁻¹(x) = (x + 6)/3 is the inverse of f(x) = 3x - 6.

Solution:

• (f ∘ f⁻¹)(x) = f(f⁻¹(x)) = f((x + 6)/3) = 3((x + 6)/3) - 6 = x + 6 - 6 = x
• (f⁻¹ ∘ f)(x) = f⁻¹(f(x)) = f⁻¹(3x - 6) = ((3x - 6) + 6)/3 = (3x)/3 = x

Since both compositions result in x, we have verified that f⁻¹(x) is indeed the inverse of f(x).

Tackling More Complex Scenarios

Worksheets often present more complex scenarios involving piecewise functions, rational functions, and functions with multiple steps. Let's address these challenges.

Piecewise Functions

Piecewise functions are defined differently over different intervals. To find the inverse, you need to find the inverse for each piece separately, ensuring you also adjust the domain and range accordingly.

Example 4:

Find the inverse of the piecewise function:

f(x) = { x² if x ≥ 0 { -x if x < 0

Solution:

For x ≥ 0: y = x² => x = √y => f⁻¹(x) = √x (for x ≥ 0) For x < 0: y = -x => x = -y => f⁻¹(x) = -x (for x > 0)

Therefore, the inverse is a piecewise function itself:

f⁻¹(x) = { √x if x ≥ 0 { -x if x > 0

Note the careful consideration of the domains to ensure consistency.

Rational Functions

Rational functions are fractions where both the numerator and denominator are polynomials. Finding their inverses often involves more intricate algebraic manipulations.

Example 5:

Find the inverse of f(x) = (2x + 1)/(x - 3)

Solution:

1. y = (2x + 1)/(x - 3)
2. x = (2y + 1)/(y - 3)
3. x(y - 3) = 2y + 1
4. xy - 3x = 2y + 1
5. xy - 2y = 3x + 1
6. y(x - 2) = 3x + 1
7. y = (3x + 1)/(x - 2)
8. f⁻¹(x) = (3x + 1)/(x - 2)

Remember to check the domain restrictions of both the original function and its inverse. The original function is undefined at x = 3, and the inverse is undefined at x = 2.

Composite Functions with Inverses

Worksheet problems often involve finding the inverse of a composite function. This requires a careful, step-by-step approach.

Example 6:

Find the inverse of h(x) = f(g(x)), where f(x) = 2x and g(x) = x + 1.

Solution:

First, find the composite function: h(x) = f(g(x)) = f(x + 1) = 2(x + 1) = 2x + 2

Now, find the inverse of h(x):

1. y = 2x + 2
2. x = 2y + 2
3. x - 2 = 2y
4. y = (x - 2)/2
5. h⁻¹(x) = (x - 2)/2

Alternatively, you could find the inverses of f(x) and g(x) individually, and then compose them in reverse order to obtain h⁻¹(x). This often simplifies the process.

f⁻¹(x) = x/2 g⁻¹(x) = x - 1 h⁻¹(x) = g⁻¹(f⁻¹(x)) = g⁻¹(x/2) = (x/2) - 1 = (x - 2)/2

This yields the same result, demonstrating the power of understanding the relationship between functions and their inverses.

Practice Makes Perfect

The key to mastering composition and inverses of functions is consistent practice. Work through numerous problems, focusing on understanding the underlying concepts and the logic behind each step. Don't just aim for the answer; aim to fully grasp the process. Review your mistakes, identify your weaknesses, and seek clarification on areas you find challenging. The more practice you get, the more confident and proficient you’ll become in solving these types of problems. Remember to always check your answers by composing the function with its inverse to ensure they result in the identity function. This verification step is crucial in confirming the accuracy of your work. By diligently practicing and applying these techniques, you’ll build a strong foundation in this essential area of mathematics.