How do I find the domain of #f(x) = 1/(sqrt(32x))#?
Let us look at some details.
There are two rules you need to keep in mind when it comes to finding the natural domain of a function. Here are the rules:
Rule 1: Nonzero in the denominator Rule 2: Nonnegative in the squareroot (eventhroot)
By Rule 1, #sqrt{32x} ne 0 Rightarrow 32x ne 0 Rightarrow 2x ne 3 Rightarrow x ne 3/2# By Rule 2, #32x ge0 Rightarrow 2x ge 3 Rightarrow x le 3/2#
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To find the domain of the function ( f(x) = \frac{1}{\sqrt{3  2x}} ), follow these steps:

Identify any restrictions on the domain due to the function's characteristics. In this case, the square root function ( \sqrt{3  2x} ) has a real square root only if the expression inside the square root, ( 3  2x ), is nonnegative. Therefore, we must have ( 3  2x \geq 0 ).

Solve the inequality ( 3  2x \geq 0 ) to find the domain of the function.
[ 3  2x \geq 0 ] [ 2x \geq 3 ] [ x \leq \frac{3}{2} ]

The domain of the function ( f(x) = \frac{1}{\sqrt{3  2x}} ) is all real numbers ( x ) such that ( x \leq \frac{3}{2} ), since this ensures that the expression inside the square root is nonnegative.
Therefore, the domain of the function is ( x \leq \frac{3}{2} ) or ([\infty, \frac{3}{2}]).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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