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Question 204714: Can you help me solve this Exponential Functions question:
1) state the domain and range of each equation
Equation 1: y = 3^ (x+4) - 1
Equation 2: y = 2^ (x -2) + 3
(everything in the brackets are exponents)
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! The domain is the set of possible x-values. When the domain is not specified (and you have to determine what it is):- Start by assuming that the domain is all Real numbers.
- Eliminate x-values you have to prevent. There are a variety of situations which must be avoided:
- Zeros in denominators.
- Negative radicands (expressions in a radical) for even-numbered roots (like square roots).
- Zero or negative expressions for the argument in a logarithm.
- Other undefined expressions like
Since neither of your functions have denominators, even-numbered roots, logarithms, etc., there is nothing to avoid. (Your functions have x in the exponent and exponents can be any number.) The domain for both is all Real numbers.
The range of a function is the set of possible y-values given the domain of the function. The range of a function is determined by examining what values the function can have for all the different possible x-values.
Since both functions have a domain of all Real numbers we have to look at each function and try to figure out what set of numbers y could have when x can be any Real number. This process requires some understanding of "how things like fractions, exponents, square roots, logarithms, etc. work".
Let's look at the first function:
The only place "x" is found is in an exponent. If we think about how exponents work we should realize that 3 to any power will- Never be negative
- Never be zero (although it can get very, very close to zero when the exponent is a large negative number.
Think about the above and try to make sense out of it. Think about negative exponents.  which is a very small fraction.
So we know that the power of 3 in the function will never get as low as zero (although it can get very close to zero). Then the function will take the power of 3 and subtract 1. So the function as a whole will never get as low as -1. On the other hand powers of 3 can get infinitely large and so, even after subtracting 1, the function will get infintely large. Putting this together, the lowest the function can be is a tiny bit above -1 and and there is no upper limit. The range is therefore, in interval notation: (-1, infinity) (which means "all numbers between -1 and infinity, not including (that's what the parentheses mean) -1 or infinity)).
Similar logic can be applied to the second function. The lowest a power of 2 can be is a tiny fraction above zero. After the function adds 3 to the power of 2 the lowest the function can be is a tiny fraction above 3. The highest a power of 2 can be is infinity. So the range of the second function is (3, infinity) (which means all numbers between 3 and infinity not including 3 and infinity).
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