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 g(x) = e^x-1

Verified step by step guidance

1. <Start by identifying the base function, which is $f(x) = e^x$. This is an exponential function with a horizontal asymptote at $y = 0$.>
2. <Recognize the transformation applied to $f(x)$. The function $g(x) = e^{x-1}$ represents a horizontal shift of the graph of $f(x) = e^x$ to the right by 1 unit.>
3. <Determine the new equation of the asymptote. Since the transformation is a horizontal shift, the horizontal asymptote remains unchanged at $y = 0$.>
4. <Identify the domain and range of the transformed function $g(x) = e^{x-1}$. The domain is all real numbers, $(-\infty, \infty)$, and the range is $(0, \infty)$, as the exponential function never reaches zero.>
5. <Use a graphing utility to confirm the transformation. Plot both $f(x) = e^x$ and $g(x) = e^{x-1}$ to observe the shift and verify the domain, range, and asymptote.>

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 'e' (approximately 2.718) is used. 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, reflecting, stretching, or compressing the graph of a function. Common transformations include vertical shifts (adding or subtracting a constant), horizontal shifts (adding or subtracting from the input), and reflections (flipping the graph over an axis). For the function g(x) = e^(x) - 1, the graph of f(x) = e^x is shifted down by 1 unit, affecting its asymptote and range.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. 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 g(x) = e^x - 1, the horizontal asymptote is y = -1, indicating that as x approaches negative infinity, the function approaches this line.