Question

In Exercises \(1-4,\) graph the function. State its domain and range. $$y=-2^{x}+3$$

Short answer

The domain of \(y=-2^{x}+3\) is \(-\infty < x < +\infty\) and the range is \(-\infty < y < 3\). The graph is an inverted exponential curve that has been shifted upwards by three units.

Step-by-step solution

Understanding the function

To start, we need to understand the function \(y=-2^{x}+3\). It is an exponential function since it has a variable as the exponent. It demonstrates vertical and horizontal transformations of the basic exponential function \(2^{x}\). The negative sign inverts the basic exponential function, making it decrease as \(x\) increases, while the '+3' vertically shifts it three units up.

Plotting points

Plot some points to create an accurate graph. For instance, use values \(x=-1, 0, 1, 2\). For each \(x\) value, calculate the corresponding \(y\) value. When \(x=-1\), \(y=-2^{-1}+3= -0.5+3=2.5\). When \(x=0\), \(y=-2^{0}+3= -1+3=2\). When \(x=1\), \(y=-2^{1}+3= -2+3=1\). And When \(x=2\), \(y=-2^{2}+3= -4+3=-1\).

Graph the function

Plot the points figured out in Step 2 and sketch the curve by connecting the points. The curve should decrease and approach the horizontal line \(y = 3\) as \(x\) goes to negative infinity, and fall towards negative infinity as \(x\) goes to positive infinity.

Identify the domain and range

The domain of an exponential function is always all real numbers, or \(-\infty < x < +\infty\). As \(x\) goes to negative infinity, \(y\) will approach 3, and as \(x\) goes to positive infinity, \(y\) will plunge to negative infinity. Thus, the range of this function is \(-\infty < y < 3\).