document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #153071 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
f(x) = 2f^x-12+30
\n" ); document.write( "Is this: $\"f%28x%29+=+2f%5E%28x-12%29+%2B+30\"$ ? 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.

\n" ); document.write( "If the second \"f\" was a typo the function is:
\n" ); document.write( " $\"f%28x%29+=+2%5E%28x-12%29+%2B+30\"$
\n" ); document.write( "then we answer the problem:
\n" ); document.write( "(1)The range is (-infinity,infinity).
\n" ); document.write( "The range is the set of possible value for the function. So we need to figure out the possible values for $\"2%5E%28x-12%29+%2B+30\"$ . If we understand exponents then we will realize that:

2 to any power can never be zero. Therefore $\"2%5E%28x-12%29+%2B+30\"$ never have a value of 30
2 to any power can never be negative. Therefore $\"2%5E%28x-12%29+%2B+30\"$ 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 $\"2%5E%28x-12%29+%2B+30\"$ 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).

\n" ); document.write( "(2) The domain is [0,infinity).
\n" ); document.write( "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)

\n" ); document.write( "(3) There is a vertical asymptote at x=12.
\n" ); document.write( "If we had a vertical asymptote at x=12 then 12 would be excluded from the domain. But 12 is in the domain.

\n" ); document.write( "(4)There is a horizontal asymptote at y=30.
\n" ); document.write( "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, $\"2%5E%28x-12%29+%2B+30\"$ becomes a very large positive number. The larger x gets, the larger $\"2%5E%28x-12%29+%2B+30\"$ gets.
\n" ); document.write( "But when x is a very large negative number, $\"2%5E%28x-12%29\"$ becomes a very tiny fraction and $\"2%5E%28x-12%29+%2B+30\"$ becomes a number just a tiny bit above 30. The more negative x gets, the tinier the fraction and the closer $\"2%5E%28x-12%29+%2B+30\"$ gets to 30. So on the left side of the graph of $\"2%5E%28x-12%29+%2B+30\"$ (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. \n" ); document.write( "

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