Question 204714
The domain is the set of possible x-values. When the domain is not specified (and you have to determine what it is):<ol><li>Start by assuming that the domain is all Real numbers.</li><li>Eliminate x-values you have to prevent. There are a variety of situations which must be avoided:<ul><li>Zeros in denominators.</li><li>Negative radicands (expressions in a radical) for even-numbered roots (like square roots).</li><li>Zero or negative expressions for the argument in a logarithm.</li><li>Other undefined expressions like {{{tan(pi/2)}}}</li></ul></li></ol>
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.<br>
The range of a function is the set of possible y-values <b>given the domain</b> 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.<br>
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".<br>
Let's look at the first function:
{{{y = 3^(x+4) - 1}}}
The only place "x" is found is in an exponent. If we think about how exponents work we should realize that 3 to <b>any</b> power will<ul><li>Never be negative</li><li>Never be zero (although it can get very, very close to zero when the exponent is a large negative number.</li></ul>Think about the above and try to make sense out of it. Think about negative exponents. {{{3^(-2000) = 1/3^2000}}} which is a <b>very</b> small fraction.<br>
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)).<br>
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).