Question

In Exercises \(1-4,\) graph the function. State its domain and range. $$y=-2^{-x}-1$$

Short answer

The graph of the function \(y = -2^{-x} - 1\) is an exponential curve reflected in the x-axis and shifted down by 1 unit. The domain of the function is all real numbers, or \(-\infty < x < \infty\), and the range is \(y < -1\).

Step-by-step solution

Understand the Function

This function is an exponential function in the form \(y = -a^{-x} - b\). In general, an exponential function is a function in which the variable is an exponent. The base of this function is 2, and there is a negative sign in front of the base which would reflect the graph in the x-axis. The number -1 at the end shifts the graph down by 1 unit.

Graph the Function

To graph the function, plot a few important points. Choose a some negative and positive x-values, calculate the corresponding y-values and plot these points in a coordinate system. Because of the nature of exponential functions, the graph will have this general shape: it will decrease towards the x-axis for positive x-values and increase immensely for negative x-values.

Identify the Domain

The domain for an exponential function is all real numbers, or \(-\infty < x < \infty\). This is because x can be any real number in an exponential function.

Identify the Range

The range for this specific exponential function is \(y < -1\). This is due to the -1 subtracted at the end of the function which shifts the graph down by 1 unit. Consequently, all y-values are less than -1.