Question

Consider the functions f(x) = (4/5)^x and g(x) = (4/5)^x + 6. What are the ranges of the two functions?

Answer 1

Answers:
The number 0 goes in the first blank for f(x)
The number 6 goes in the second blank for g(x)

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Explanation:

As x gets larger and larger, the expression (4/5)^x will get smaller and smaller. What's going on is that we are repeatedly multiplying (4/5) with itself over and over if we assume x is some whole number. For example, if x = 100, then we'll have 100 copies of (4/5) multiplied out. Each time we multiply by (4/5), the result gets smaller. 

The results will approach 0 but not actually get there. So this is why f(x) has the horizontal asymptote y = 0. The function values f(x) will essentially be the set of positive y values, in other words, y > 0. This covers the range for f(x). To get the range of g(x), we add 6 to the range of f(x). This is because everything on g(x) is a vertical shift upward of 6 units.