Problem 74 - Calculus

Question

Give an example of: An increasing function \(f(x)\) whose values are greater than those of its inverse function \(f^{-1}(x)\) for \(x>0\).

Step-by-Step Solution

Answer

The function \( f(x) = e^x \) satisfies \( f(x) > f^{-1}(x) \) for \( x > 0 \).

Step 1: Understanding the Problem

The task is to find an increasing function \( f(x) \) such that \( f(x) > f^{-1}(x) \) for \( x > 0 \). An increasing function is one where if \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \). We also need to confirm that the function is invertible, meaning every \( x \) in the domain maps to a unique \( f(x) \).

Step 2: Choosing a Function

Let's consider the function \( f(x) = e^x \), which is continuous and strictly increasing for all values of \( x \). It is also invertible since it has an inverse function \( f^{-1}(x) = \ln(x) \).

Step 3: Verifying the Function

For the chosen function \( f(x) = e^x \) and its inverse \( f^{-1}(x) = \ln(x) \), we need to check if \( e^x > \ln(x) \) holds for \( x > 0 \).

Step 4: Testing the Inequality

The function \( e^x \) grows exponentially while \( \ln(x) \) grows slowly as a logarithmic function. At small positive values of \( x \), such as \( x = 1 \), we have \( e^1 = e \approx 2.718 \) and \( \ln(1) = 0 \), so \( e > 0 \). As \( x \) increases, \( e^x \) remains greater than \( \ln(x) \).

Step 5: Drawing Conclusion

Because \( e^x > \ln(x) \) holds for all \( x > 0 \) and \( e^x \) is an increasing function, the function \( f(x) = e^x \) satisfies the problem's condition that \( f(x) > f^{-1}(x) \) for \( x > 0 \).

Key Concepts

Invertible Function

In mathematics, an invertible function, or bijective function, is a key concept that ensures each output value comes from one and only one input value. This property is crucial for a function to have an inverse. Unlike simple functions that might map several inputs to one output, a function that is invertible has a unique counterpart for every element in its range, allowing us to perfectly "reverse" the function through its inverse.

Here's what makes a function invertible:

• Bijectiveness: The function must be both injective (no two distinct inputs have the same output) and surjective (every possible output is accounted for).
• One-to-One Correspondence: Each element in the function's domain maps to a unique element in the range.

For example, the exponential function \( f(x) = e^x \) is invertible because it is strictly increasing and every real number maps to a unique positive number. This makes it possible for its inverse, the natural logarithm function \( f^{-1}(x) = \ln(x) \), to "undo" \( e^x \), returning you to the original input.

Exponential Function

An exponential function is a type of function where the variable is in the exponent, usually in the form \( f(x) = a^x \), where \( a \) is a constant. These functions are characterized by their rapid growth, making them very powerful in representing processes that increase quickly over time, such as compound interest or population growth.

Some key properties of exponential functions include:

• Rapid Increase: For \( a > 1 \), as \( x \) increases, \( a^x \) increases much faster than any linear or polynomial function.
• Base of Natural Exponential Function: The exponential function \( e^x \), where \( e \approx 2.718 \), is especially important in calculus and natural sciences due to its unique derivative properties. The slope of \( e^x \) at any point \( x \) is exactly \( e^x \).
• Strictly Increasing or Decreasing: If \( a > 1 \), \( a^x \) is strictly increasing. If \( 0 < a < 1 \), the function is decreasing.

Exponential functions like \( f(x) = e^x \) play a crucial role in several fields because they model growth processes effectively and are easily reversible, providing the basis for logarithmic functions.

Logarithmic Function

Logarithmic functions are the inverses of exponential functions and are written in the form \( g(x) = \log_a(x) \), where \( a \) is the base of the logarithm. These functions convert multiplicative operations into additive ones, which is incredibly useful for dealing with large numbers.

Here's how logarithmic functions work:

• Inverse Function: If \( a^x = b \), then \( x = \log_a(b) \). This property makes logarithms essential for solving equations involving exponents.
• The Natural Logarithm: The most common base is \( e \), leading to the natural logarithm \( \ln(x) \), which is prevalent in calculus and scientific computations due to its various mathematical properties.
• Simplifies Calculations: Logarithms make complex calculations easier, especially in fields like acoustics or earthquake measurement, where they help convert multiplicative processes into more manageable additions.

Logarithmic functions grow much slower compared to their exponential counterparts, which is beneficial for certain analyses and comparisons, such as determining rates of change or solving for time in exponential growth models.