Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=e^x, g(x)=e^x-1\)

Short answer

The function \(g(x) = e^x - 1\) is a downward shift by 1 unit of the original function \(f(x) = e^x\). The graph of \(f(x) = e^x\) crosses the y-axis at (0,1), while the graph of \(g(x) = e^x - 1\) crosses the y-axis at (0,0).

Step-by-step solution

Understand the Functions

The given function \(f(x) = e^x\) is an exponential function where \(e\) is the base. The function \(g(x) = e^x - 1\) is a transformation of the given function \(f(x) = e^x\).

Identify the Transformation

The function \(g(x) = e^x - 1\) is ↓1 units shift of the original function \(f(x) = e^x\).

Graph the Original Function

Graph the function \(f(x) = e^x\) on x-y plane. This function crosses the y-axis at (0,1) and increases exponentially for positive x values, while approaching 0 for negative x values.

Graph the Transformed Function

To graph \(g(x) = e^x - 1\), shift the graph of \(f(x) = e^x\) downward by 1 unit. This graph will cross the y-axis at (0,0), increase exponentially for positive x values, and approach -1 for negative x values.