Begin by graphing f(x) = 2^x. Then use transformations of this graph to graph the given function. Be sure to graph and 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. h(x) = 2^x+1 – 1

Verified step by step guidance

Start by understanding the base function: \(f(x) = 2^x\). This is an exponential function with a horizontal asymptote at \(y = 0\), domain \((-\infty, \infty)\), and range \((0, \infty)\).

Identify the transformations in the given function \(h(x) = 2^{x+1} - 1\). The term \(x + 1\) inside the exponent indicates a horizontal shift to the left by 1 unit.

The \(-1\) outside the exponential function represents a vertical shift downward by 1 unit. This will move the entire graph down by 1.

Determine the new equation of the asymptote after the vertical shift. Since the original asymptote is \(y = 0\), shifting down by 1 changes it to \(y = -1\).

Using these transformations, sketch the graph of \(h(x)\) by shifting the graph of \(f(x)\) left by 1 unit and down by 1 unit. Then, state the domain as \((-\infty, \infty)\) and the range as \((-1, \infty)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. The graph of f(x) = 2^x is a smooth curve increasing rapidly, passing through (0,1), and is fundamental for understanding transformations applied to it.

Transformations of Functions

Transformations modify the graph of a base function by shifting, stretching, compressing, or reflecting it. For h(x) = 2^(x+1) - 1, the term (x+1) shifts the graph left by 1 unit, and subtracting 1 shifts it down by 1 unit, altering the position but not the shape.

Asymptotes, Domain, and Range

Exponential functions have a horizontal asymptote, typically y=0 for f(x)=2^x. Transformations shift this asymptote accordingly. The domain of exponential functions is all real numbers, while the range depends on vertical shifts; understanding these helps in graphing and interpreting the function.

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