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An exponential function is a mathematical function of the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the mathematical constant approximately equal to 2.71828, and \( a \) and \( b \) are constants. The base of the natural exponent \( e \) gives these functions unique properties. In our exercise, the function \( f(x) = 2 - e^x \) includes an exponential decay term, \( e^x \), which decreases as \( x \) increases, causing the whole function to decrease.Exponential functions are characterized by:

• Rapid growth or decay: Depending on the coefficient, they can model steep increases or decreases over short intervals.
• A horizontal asymptote: As \( x \) approaches positive infinity, the function approaches a constant value.

In our case, the function \( 2 - e^x \) has a horizontal asymptote at \( y = 2 \). As \( x \) becomes very large, \( e^x \) tends to zero, making the function approach \( y = 2 \). This property impacts how the graph behaves and how it reflects when considering its inverse.

When we talk about graph reflection, we mean flipping a graph over a specific line so that each point on the graph has a symmetrical counterpart. For inverse functions, a common line of reflection is \( y = x \). If a function and its inverse are plotted, ideally, they should appear as mirror images relative to this line.In our example, we have:

• The original function \( f(x) = 2 - e^x \).
• Its inverse \( f^{-1}(x) = \ln(2 - x) \).
• The line \( y = x \) which serves as the mirror line for reflection.

To check if the graphs truly exhibit this reflective property:

• Select a point on the function's graph. For \( f(x) = 2 - e^x \), this could be the point \((0, 1)\).
• Locate where this point reflects on the inverse graph \( f^{-1}(x) = \ln(2 - x) \). You should find that if \((a, b)\) is on \( f \), then \((b, a)\) is on \( f^{-1} \).
• If both points are indeed symmetrical about the line \( y = x \), reflection is confirmed.

This property is essential for verifying if two functions are truly inverses of each other.

The natural logarithm, denoted by \( \ln(x) \), is the inverse operation of exponentiation with base \( e \). While an exponential function raises \( e \) to a power, the natural logarithm finds which power an exponential function was raised to get a certain number.Consider the relation in finding the inverse:

• We begin with \( e^x = 2 - y \).
• Using \( \ln \) on both sides \( x = \ln(2 - y) \), reveals how \( x \) is derived from the exponential term \( 2 - y \).

The natural logarithm has key properties:

• \( \ln(1) = 0 \).
• \( \ln(e) = 1 \). The natural log "undoes" the exponent \( e \), reinforcing its role as an inverse.
• It increases logarithmically (slowly at first), as input values increase from a positive number.

In our exercise, finding \( f^{-1}(x) = \ln(2 - x) \) applies this concept to express the inverse of the exponential decay in a way that allows easy graphing and understanding of the function's behavior.