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-1)+2

Verified step by step guidance

1. Identify the base function: The base function is \( f(x) = e^x \).
2. Apply the horizontal shift: The function \( h(x) = e^{(x-1)} + 2 \) indicates a horizontal shift to the right by 1 unit, resulting in \( e^{x-1} \).
3. Apply the vertical shift: The \(+2\) outside the exponential function indicates a vertical shift upwards by 2 units.
4. Determine the asymptote: The horizontal asymptote of \( f(x) = e^x \) is \( y = 0 \). After the vertical shift, the new asymptote is \( y = 2 \).
5. Determine the domain and range: The domain of \( h(x) = e^{(x-1)} + 2 \) is all real numbers \((-\infty, \infty)\), and the range is \((2, \infty)\) due to the vertical shift.

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 are characterized by their rapid growth or decay and have unique properties such as a horizontal asymptote.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, in the function h(x) = e^(x-1) + 2, the graph of f(x) = e^x is shifted right by 1 unit and up by 2 units. Understanding these transformations helps in predicting how the graph will change without having to plot every point.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For exponential functions, the horizontal asymptote is typically found at y = k, where k is a constant that represents the vertical shift of the function. In the case of h(x) = e^(x-1) + 2, the horizontal asymptote is y = 2, indicating that as x approaches negative infinity, the function values approach 2.