Graph the given function.
$g\left(x\right)=4^{-x}-1$

• A.
• B.
• C.
• D.

Answer: (B)

Verified step by step guidance

1. Identify the function type: The given function is an exponential function of the form g(x) = 4^{-x} - 1.
2. Determine the horizontal asymptote: For the function g(x) = 4^{-x} - 1, the horizontal asymptote is y = -1, since the term -1 shifts the graph of 4^{-x} down by 1 unit.
3. Analyze the behavior of the function: As x approaches positive infinity, 4^{-x} approaches 0, so g(x) approaches -1. As x approaches negative infinity, 4^{-x} becomes very large, so g(x) increases without bound.
4. Identify key points: Calculate a few key points to help sketch the graph. For example, when x = 0, g(x) = 4^{0} - 1 = 0. When x = 1, g(x) = 4^{-1} - 1 = -0.75. When x = -1, g(x) = 4^{1} - 1 = 3.
5. Sketch the graph: Plot the key points and draw the curve approaching the horizontal asymptote y = -1 as x increases, and rising steeply as x decreases. The graph should pass through the calculated points and reflect the behavior of an exponential decay function.