Which Of The Following Pairs Are Inverses Of Each Other

Let's explore the concept of inverse functions and how to determine if two given functions are inverses of each other. Understanding inverses is fundamental in mathematics, especially when dealing with algebraic manipulations, solving equations, and analyzing function behavior. This article will provide a comprehensive guide, complete with examples and explanations.

Understanding Inverse Functions

In mathematics, an inverse function is a function that "reverses" another function. If a function f takes x to y, then the inverse function, denoted as f⁻¹, takes y back to x. In other words, f(x) = y if and only if f⁻¹(y) = x.

Key Characteristics of Inverse Functions

• Reversal of Operations: Inverse functions undo the operations of the original function.
• Domain and Range Swap: The domain of f becomes the range of f⁻¹, and vice versa.
• Composition Property: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This property is crucial for verifying if two functions are inverses.
• Graphical Symmetry: The graphs of f and f⁻¹ are symmetric with respect to the line y = x.

Criteria for Determining Inverse Functions

To determine if two functions, f(x) and g(x), are inverses of each other, the following condition must be satisfied:

• f(g(x)) = x for all x in the domain of g, AND
• g(f(x)) = x for all x in the domain of f.

If both conditions hold true, then f(x) and g(x) are indeed inverses. If either condition fails, they are not inverses.

Step-by-Step Verification Process

Here’s a detailed process to verify if two functions are inverses:

1. Compose f(g(x)): Substitute g(x) into f(x) wherever you see x. Simplify the expression.
2. Compose g(f(x)): Substitute f(x) into g(x) wherever you see x. Simplify the expression.
3. Check if f(g(x)) = x and g(f(x)) = x: Verify that both composite functions simplify to x.
4. Consider the Domains: Ensure that the compositions are valid for all x in the respective domains.

Examples of Determining Inverse Functions

Let's walk through several examples to illustrate the process of verifying inverse functions.

Example 1: Linear Functions

Consider the functions:

• f(x) = 2x + 3
• g(x) = (x - 3) / 2

Step 1: Compose f(g(x)):

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

Step 2: Compose g(f(x)):

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

Step 3: Check if f(g(x)) = x and g(f(x)) = x:

Both compositions equal x.

Conclusion:

Since both f(g(x)) = x and g(f(x)) = x, the functions f(x) = 2x + 3 and g(x) = (x - 3) / 2 are inverses of each other.

Example 2: Quadratic and Square Root Functions

Consider the functions (for x ≥ 0):

• f(x) = x²
• g(x) = √x

Step 1: Compose f(g(x)):

f(g(x)) = f(√x) = (√x)² = x

Step 2: Compose g(f(x)):

g(f(x)) = g(x²) = √(x²) = |x|

Since we are given the condition x ≥ 0, then |x| = x.

Step 3: Check if f(g(x)) = x and g(f(x)) = x:

Both compositions equal x under the condition x ≥ 0.

Conclusion:

Since both f(g(x)) = x and g(f(x)) = x for x ≥ 0, the functions f(x) = x² and g(x) = √x are inverses of each other on the restricted domain.

Example 3: Rational Functions

Consider the functions:

• f(x) = (x + 1) / (x - 2)
• g(x) = (2x + 1) / (x - 1)

Step 1: Compose f(g(x)):

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

To simplify, multiply the numerator and denominator by (x - 1):

f(g(x)) = ((2x + 1) + (x - 1)) / ((2x + 1) - 2(x - 1)) = (3x) / (2x + 1 - 2x + 2) = (3x) / 3 = x

Step 2: Compose g(f(x)):

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

To simplify, multiply the numerator and denominator by (x - 2):

g(f(x)) = (2(x + 1) + (x - 2)) / ((x + 1) - (x - 2)) = (2x + 2 + x - 2) / (x + 1 - x + 2) = (3x) / 3 = x

Step 3: Check if f(g(x)) = x and g(f(x)) = x:

Both compositions equal x.

Conclusion:

Since both f(g(x)) = x and g(f(x)) = x, the functions f(x) = (x + 1) / (x - 2) and g(x) = (2x + 1) / (x - 1) are inverses of each other.

Example 4: Exponential and Logarithmic Functions

Consider the functions:

• f(x) = eˣ
• g(x) = ln(x)

Step 1: Compose f(g(x)):

f(g(x)) = f(ln(x)) = e^(ln(x)) = x

Step 2: Compose g(f(x)):

g(f(x)) = g(eˣ) = ln(eˣ) = x

Step 3: Check if f(g(x)) = x and g(f(x)) = x:

Both compositions equal x.

Conclusion:

Since both f(g(x)) = x and g(f(x)) = x, the functions f(x) = eˣ and g(x) = ln(x) are inverses of each other.

Example 5: Functions that are NOT Inverses

Consider the functions:

• f(x) = x³ + 1
• g(x) = √(x - 1) (Note: This should actually be the cube root, but let's proceed with the square root to show non-inverses)

Step 1: Compose f(g(x)):

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

Step 2: Compose g(f(x)):

g(f(x)) = g(x³ + 1) = √((x³ + 1) - 1) = √(x³)

Step 3: Check if f(g(x)) = x and g(f(x)) = x:

f(g(x)) = (x - 1)^(3/2) + 1 which is not equal to x. g(f(x)) = √(x³) which is not equal to x.

Conclusion:

Since neither f(g(x)) nor g(f(x)) simplifies to x, the functions f(x) = x³ + 1 and g(x) = √(x - 1) are not inverses of each other.

Importance of Domain and Range

When determining if two functions are inverses, it is crucial to consider the domain and range of both functions. The domain of f must match the range of f⁻¹, and the range of f must match the domain of f⁻¹. Restrictions on the domain may be necessary to ensure that the inverse function exists.

Example: Restricted Domain

Consider the function f(x) = x². Without any restrictions, this function does not have an inverse over the entire real number line because it fails the horizontal line test. However, if we restrict the domain to x ≥ 0, then the inverse function f⁻¹(x) = √x is well-defined.

Graphical Interpretation

The graphs of a function and its inverse are symmetric with respect to the line y = x. This means that if you were to fold the graph along the line y = x, the function and its inverse would overlap.

Verifying Inverses Graphically

1. Plot the functions f(x) and g(x) on the same coordinate plane.
2. Draw the line y = x.
3. Observe if the graphs of f(x) and g(x) are symmetric about the line y = x.

If the graphs are symmetric, then the functions are likely inverses. However, graphical verification is not as precise as algebraic verification, so it should be used in conjunction with the composition method.

Common Mistakes to Avoid

• Assuming that 1/f(x) is the inverse of f(x): This is a common mistake. The inverse function f⁻¹(x) is not the same as the reciprocal function 1/f(x).
• Forgetting to check both f(g(x)) and g(f(x)): Both compositions must equal x for the functions to be inverses.
• Ignoring domain restrictions: Domain restrictions can significantly affect whether two functions are inverses.
• Incorrectly simplifying composite functions: Be careful with algebraic manipulations when simplifying f(g(x)) and g(f(x)).

Advanced Examples

Trigonometric Functions and Their Inverses

Trigonometric functions (such as sine, cosine, and tangent) have inverses, but these inverses are typically defined on restricted domains to ensure they are functions (i.e., they pass the vertical line test).

For example:

• f(x) = sin(x), defined on [-π/2, π/2]
• g(x) = arcsin(x) or sin⁻¹(x), defined on [-1, 1]

Here, f(g(x)) = sin(arcsin(x)) = x and g(f(x)) = arcsin(sin(x)) = x for x in their respective domains.

Hyperbolic Functions and Their Inverses

Hyperbolic functions also have inverses. For example:

• f(x) = sinh(x) = (eˣ - e⁻ˣ) / 2
• g(x) = arcsinh(x) = ln(x + √(x² + 1))

Again, f(g(x)) = sinh(arcsinh(x)) = x and g(f(x)) = arcsinh(sinh(x)) = x for all real numbers.

Practical Applications

Understanding inverse functions is crucial in various fields:

• Cryptography: Inverse functions are used in encryption and decryption processes.
• Engineering: They are used in control systems and signal processing.
• Computer Graphics: Inverse transformations are used in rendering and animation.
• Calculus: Inverses play a significant role in integration techniques.

Conclusion

Determining whether two functions are inverses of each other involves verifying that their compositions, f(g(x)) and g(f(x)), both simplify to x. Additionally, it's vital to consider the domain and range of each function. By following the outlined steps and understanding the underlying principles, you can confidently identify inverse function pairs. Through examples ranging from simple linear functions to more complex rational, exponential, and trigonometric functions, this guide has equipped you with the knowledge to tackle various inverse function problems.