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First, let us be clear that what we're looking at here is an exponential function.
That said, we can use transformations to determine the graph of said function by looking at its parent function, f(x) = 2^x.
Since our function is f(x) = 2^(-x) - 2, it is a transformation of the parent function by a reflection across the y-axis and a vertical shift of -2. This can be demonstrated on any standard graphing calculator by graphing y1 = 2^x, y2 = 2^(-x) - 2 and comparing the graphs of the two.
If you consider the equation f(x) = 2^(-x), it is clear that this function will have a horizontal asymptote at y=0, as for larger values of x, the value of f(x) will approach 0. Thus, for our function, f(x) = 2^(-x) - 2, it is clear that it will have a horizontal asymptote at y = -2. This can be affirmed either graphically or through looking a table of the function for large values of x.
From this we can determine that the range of f(x) is all real numbers greater than -2. Since there are no conditions that will result in division by zero over the function, the domain of f(x) is simply all real numbers.
You need to find what are the transformations suffered by the original function such that the new transformed function is `f(x)=2^(-x) - 2` .
You need to remember that if `x` is replaced by `-x` in equation of function it means that the graph of the original function `y=f(x)` is reflected through y axis.
You should consider the original function `y = 2^x` , hence the graph of `y = 2^(-x)` is obtained reflecting the graph of `y = 2^x` through y axis. The red graph represents the graph of `y=2^(-x)` and you should notice that you may obtain this graph mirroring the black graph through y axis.
Notice the presence of negative constant term -2 that represents a vertical down translation of the graph `y = 2^(-x)` such that:
The red curve represents the graph of `f(x)=2^(-x)- 2`
You need to remember that the domain of exponential function is the set R.
You need to solve for x the equation `2^(-x)- 2 = 0 =gt 2^(-x) = 2 =gt x = -1` , then you may find the range of the function, hence y in `(0;oo)` .
Hence, the domain of transformed function is R and the range is `(0,oo) ` .
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