How do you find the domain and range of #f(x) = sqrt(4+x) / (1x)#?
Domain:
Range:
This suggests that you must
Consequently, you could say that the function's domain will be
This indicates that the function's range is
sqrt(4+x)/(1x) [10, 10, 5, 5]} is the graph.
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To find the domain and range of the function ( f(x) = \frac{\sqrt{4+x}}{1x} ):

Domain: Set the denominator equal to zero and solve for ( x ) to find any excluded values. Then determine if there are any additional restrictions on ( x ) from the square root expression.
Solving ( 1  x = 0 ) gives ( x = 1 ). Thus, ( x = 1 ) is excluded from the domain due to division by zero.
Additionally, since we have a square root expression, the expression under the square root (( 4 + x )) must be nonnegative: [ 4 + x \geq 0 ] Solving this inequality gives ( x \geq 4 ).
Therefore, the domain of the function is ( x \in (\infty, 1) \cup (1, \infty) ).

Range: Analyze the behavior of the function as ( x ) approaches its domain boundaries to determine the range.
As ( x ) approaches ( \infty ), both numerator and denominator tend toward positive infinity, so ( f(x) ) approaches zero.
As ( x ) approaches 1 from the left (( x < 1 )), the expression ( \sqrt{4 + x} ) approaches ( \sqrt{4} = 2 ) and the denominator approaches ( 0^+ ), so ( f(x) ) approaches positive infinity.
As ( x ) approaches 1 from the right (( x > 1 )), the expression ( \sqrt{4 + x} ) approaches ( \sqrt{4} = 2 ) and the denominator approaches ( 0^ ), so ( f(x) ) approaches negative infinity.
Hence, the range of the function is ( f(x) \in (\infty, 0) \cup (0, \infty) ).
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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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