We want to find the range of the function f(x)=- ab^x where a>0 and b>1. Let's first consider the general form of an exponential function.
f(x)=ab^x
General Form of an Exponential Function
An exponential function is defined when a≠0, b>0, and b≠1. If a>0, the range of the function is all real numbers greater than 0 for all possible values of b.
Range: f(x) > 0Note that 0 is not included in the range because y=0 is the asymptote. To illustrate, let's look at an example of an exponential function where b is greater than 1.
f(x)=a* 2^x
Let's now examine how the graph of this function changes as the values of a change to be various values greater than 0.
As we can see from the graph, the range of the function in the form f(x)=ab^x is always all positive real numbers when a>0 and b>1.
Given Exponential Function
Let's now consider the given function.
f(x)=- ab^x
Notice that this function has a negative sign in front of a. This reflects the graph of the general function f(x)=ab^x across the x-axis.
Let's take a look at an example where b is equal to 2, just as we did with the general form.
f(x)=- a* 2^x
Now we can examine how the graph of this function changes for values of a that are greater than 0.
This graph shows that the range of the function in the form f(x)=- ab^x is all negative real numbers. Once again, 0 is not included in the range because y=0 is the asymptote of the graph — as the function gets closer and closer to y=0, it will never touch it.
Range: f(x) < 0
