The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e^-x

Verified step by step guidance

1. Identify the base function: The base function is \( f(x) = e^x \).
2. Determine the transformation: The function \( h(x) = e^{-x} \) is a reflection of \( f(x) = e^x \) across the y-axis.
3. Graph the transformation: Reflect the graph of \( f(x) = e^x \) across the y-axis to obtain the graph of \( h(x) = e^{-x} \).
4. Identify the asymptote: The horizontal asymptote of \( h(x) = e^{-x} \) is the same as \( f(x) = e^x \), which is \( y = 0 \).
5. Determine the domain and range: The domain of \( h(x) = e^{-x} \) is all real numbers \(( -\infty, \infty )\), and the range is \(( 0, \infty )\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. The function f(x) = e^x is a specific case where the base 'b' is Euler's number (approximately 2.718). These functions exhibit rapid growth or decay, depending on the value of 'b', and are characterized by their continuous and smooth curves.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function h(x) = e^-x represents a reflection of f(x) = e^x across the y-axis. Understanding these transformations allows one to manipulate the original graph to obtain the graph of a new function while maintaining the same general shape.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. For exponential functions, the horizontal asymptote is typically found at y = 0, indicating that as x approaches negative infinity, the function's value approaches zero. Identifying asymptotes is crucial for understanding the behavior of the graph, particularly in determining the domain and range of the function.