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Question 202899: Which one of the following is true about the graph of f(x)=2f^x-12+30? (1)The range is (-infinity,infinity). (2) The domain is bracket0,infinity). (3) There is a vertical asymptote at x=12. (4)There is a horizontal asymptote at y=30. Thank-you for your help, very much appriciated.
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! f(x) = 2f^x-12+30
Is this:  ? If so:- It is very unusual (and confusing) for "f" to be used both as the name of the function and as a variable!
- The function "f" is not a function of just "x". It is a function of both "x" and the variable "f"! It would be f(x, f).
- Please correct the problem and repost it. Please use more parentheses to ensure clarity.
If the second "f" was a typo the function is:
then we answer the problem:
(1)The range is (-infinity,infinity).
The range is the set of possible value for the function. So we need to figure out the possible values for  . If we understand exponents then we will realize that:- 2 to any power can never be zero. Therefore never have a value of 30
- 2 to any power can never be negative. Therefore can never be less than 30.
- 2 to a very large negative power will be a very tiny fraction which is very close to zero in value. Therefore can be very, very close to (but never equal to) 30 when x is a large negative number.
- 2 to a power will become an infinitely large positive number as x gets to be an infinitely large positive number.
Therefore the range is (30, infinity).
(2) The domain is [0,infinity).
The domain is the set of possible x-values. The x in f(x) is found only in the exponent of 2. Since exponents can be any number and since there are no other reasons to exclude x-values (like zeros in denominators, negatives in a square roots, zeros or negatives in logarithms, etc.) the domain of f(x) is all Real numbers: (-infinity, infinity)
(3) There is a vertical asymptote at x=12.
If we had a vertical asymptote at x=12 then 12 would be excluded from the domain. But 12 is in the domain.
(4)There is a horizontal asymptote at y=30.
Horizontal asymptotes occur when the function values approach a certain number when x-values become very large positive or negative numbers. When x is a very large positive number, becomes a very large positive number. The larger x gets, the larger gets.
But when x is a very large negative number,  becomes a very tiny fraction and becomes a number just a tiny bit above 30. The more negative x gets, the tinier the fraction and the closer gets to 30. So on the left side of the graph of (where x gets more and more negative, the graph will get closer and closer to y = 30. We do have a horizontal asymptote at y = 30.
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