SOLUTION: Without graphing, find the domain of the following functions:
a){{{f(x) = sqrt((x - 3)/(2x^2 - 8))}}}, and
b){{{f(x) = root(3,(x + 5)/(x^2 - 9))}}}.
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Question 389690: Without graphing, find the domain of the following functions:
a), and
b).
Found 2 solutions by josmiceli, robertb:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
a)
if
The numerator is and the denominator is (+)
and
If , The numerator is imaginary
-------------------
If , numerator is (+)
and denominator is (+)
-------------------
If
The denominator is
and the numerator is imaginary
If , the denominator is imaginary
-------------------
The restriction includes
the restriction , so that is the domain
answer
b) .
If the numerator is
and the denominator is (+)
If the numerator is (-)
and the denominator is (+)
If the numerator is (-)
and the denominator is (+) or (-)
---------------------------
If , the denominator is
If or denominator is (+)
-----------------------------
not allowed is the only restriction
The domain is and
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
a) Since there is a square root symbol, we have to ensure that . To find the critical numbers, determine all x values that make either the top or the bottom equal to 0. By inspection they are -2, 2, and 3. These critical numbers partition the real number line into 4 parts. Choosing the test numbers -3, 0, 2.5, and 4, and checking for the signs:
For (, -2), .
For (-2, 2), .
For (2,3), .
For (3, ), .
Of the critical numbers, only 3 can be accepted, and not 2 and -2. Therefore the domain is (-2, 2)U [3, ).
b) The radical symbol is a cube-root symbol, and every real number has a cube root (whether positive or negative). Then the only thing we have to ensure is that the denominator won't be 0. The x-values that make the denominator 0 are -3 and 3. Hence the domain is the set of all real numbers except -3 and 3, or R\{-3, 3}.
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