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 with the base function given: \(f(x) = e^{x}\). This is the natural exponential function with a horizontal asymptote at \(y = 0\), domain \((-\infty, \infty)\), and range \((0, \infty)\).
2. Identify the transformation in the function \(g(x) = e^{x} - 1\). Here, the graph of \(f(x)\) is shifted vertically downward by 1 unit because of the \(-1\) outside the exponential.
3. Apply the vertical shift to the asymptote. Since the original asymptote is \(y = 0\), shifting down by 1 unit changes the asymptote to \(y = -1\).
4. Determine the domain and range of \(g(x)\). The domain remains all real numbers \((-\infty, \infty)\) because the exponential function is defined for all \(x\). The range shifts down by 1, so it becomes \((-1, \infty)\).
5. To confirm your hand-drawn graph, use a graphing utility to plot \(g(x) = e^{x} - 1\) and observe the vertical shift and the new asymptote at \(y = -1\).

Key Concepts

Exponential Functions and Their Graphs

An exponential function has the form f(x) = a^x, where a > 0 and a ≠ 1. The graph of f(x) = e^x is a smooth curve increasing rapidly, passing through (0,1). Understanding this base graph is essential before applying transformations.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the base graph. For g(x) = e^(x) - 1, the graph shifts downward by 1 unit, affecting the position of the curve and its asymptote. Recognizing these changes helps in sketching and analyzing the new graph.

Domain, Range, and Asymptotes of Exponential Functions

The domain of e^x and its transformations is all real numbers, while the range depends on vertical shifts. The horizontal asymptote for e^x is y=0, but shifts like in g(x) = e^x - 1 move the asymptote to y = -1. Identifying these features is key to understanding the function's behavior.