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An exponential function is a mathematical expression where the variable is located in the exponent. This type of function is commonly written in the form \( f(x) = a \cdot e^{bx} \), where \( a \) is a constant, \( e \) is the base of natural logarithms (approximately 2.718), and \( b \) is the rate of growth or decay. The beauty of exponential functions lies in their consistent multiplicative growth or decay patterns.

Exponential functions are vital because they model a variety of real-world phenomena, such as population growth, radioactive decay, and interest calculations. The function \( f(x) = -e^{-x} \) is a specific type of exponential function where the base \( e \) is raised to a negative exponent, leading to a decay function since \( b = -1 \). This negative exponent signifies a flip across the y-axis, as well as a steady decline in value as \( x \) increases.

Function evaluation is essential when you want to find the value of a function for a specific input. Simply put, it involves substituting the input value into the function and simplifying the resulting expression.

In our example, to find the y-intercept of the function \( f(x) = -e^{-x} \), we set \( x = 0 \) because the y-intercept is the point where the graph of the function crosses the y-axis. By substituting 0 for \( x \), the function simplifies:

• \( f(0) = -e^{0} \)
• \( e^{0} = 1 \) because any number to the zero power is 1.
• Therefore, \( f(0) = -1 \).

This shows us that the y-intercept of the function is actually \((0, -1)\), contrary to the given statement that it is \((0, 1)\). Being able to evaluate a function correctly helps verify crucial points about the graph and its behavior.

Graphing functions is a crucial skill in mathematics, allowing us to visually interpret how a function behaves. For exponential functions like \( f(x) = -e^{-x} \), graphing can reveal trends such as growth or decay.

Firstly, it's important to understand the general shape of a graph for exponential functions. For \( f(x) = e^{x} \), the graph would show exponential growth. However, with \( f(x) = -e^{-x} \), two transformations occur:

• The negative sign in \( -e^{-x} \) causes a reflection across the x-axis.
• The negative exponent \( -x \) inverts the function, demonstrating decay rather than growth.

The graph will start from a point just below the x-axis at the y-intercept \((0, -1)\) and approach the x-axis as \( x \) increases, but will never cross or touch it. Observing these trends through graphing helps in understanding how different terms in the exponential function influence its overall behavior.