The figure shows the graph of f(x)=e^x . Use transformations...

The figure shows the graph of $f(x)=e^{x} .$ Use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. (Graph can't copy) $$g(x)=e^{x+1}$$

Key Concepts

Domain and Range

The domain of a function refers to all possible input values, while the range refers to all possible output values. With exponential functions, the domain is typically all real numbers, and the range is usually all positive real numbers. These characteristics often remain unchanged under horizontal translations, ensuring that the function's basic behavior is preserved even when its graph is shifted.

Asymptotes

An asymptote is a line that the graph of a function approaches but never actually touches. In the context of exponential functions, the horizontal asymptote typically represents a boundary for the function's output values. For basic exponential functions like e^x and their shifted versions, the horizontal asymptote remains y = 0, showcasing that the function values approach zero as x tends to negative infinity.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. Understanding how these transformations affect a function’s graph is crucial for interpreting and sketching functions without plotting numerous individual points. This includes both vertical and horizontal shifts, which reposition the graph relative to its original location.

Exponential Functions

Exponential functions, such as those with the form y = e^x, are functions that have a constant base raised to a variable exponent. They have distinctive properties including rapid growth or decay and a horizontal asymptote. These functions are fundamental in mathematics and appear in various contexts such as compound interest, population growth, and natural processes.

Horizontal Translations

A horizontal translation shifts the graph of a function left or right. For a function such as y = e^(x + c), replacing x with (x + c) translates the graph horizontally by –c units. This type of transformation affects the input of the function, moving each point on the graph uniformly along the x-axis.

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