Questions: Use transformations of the graph of f(x)=2^x to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x)=2^x-2 Graph g(x)=2^x-2 and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

Use transformations of the graph of f(x)=2^x to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs.
g(x)=2^x-2

Graph g(x)=2^x-2 and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

Transcript text: Use transformations of the graph of $f(x)=2^{x}$ to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. \[ g(x)=2^{x}-2 \] Graph $\mathrm{g}(\mathrm{x})=2^{\mathrm{x}}-2$ and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

Solution

Solution Steps

Step 1: Identify the Base Function

The base function given is $ f(x) = 2^x $.

Step 2: Apply the Transformation

The given function is $ g(x) = 2^x - 2 $. This represents a vertical shift of the base function $ f(x) = 2^x $ downward by 2 units.

Step 3: Determine the Asymptote

For the base function $ f(x) = 2^x $, the horizontal asymptote is $ y = 0 $. After the vertical shift downward by 2 units, the new horizontal asymptote is $ y = -2 $.

Final Answer

The function $ g(x) = 2^x - 2 $ is a vertical shift of $ f(x) = 2^x $ downward by 2 units.
The horizontal asymptote of $ g(x) $ is $ y = -2 $.
The domain of $ g(x) $ is $ (-\infty, \infty) $.
The range of $ g(x) $ is $ (-2, \infty) $.

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