# How do you find the domain and range of #1/(x+1)+3#?

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be.

To find any excluded value in the range, rearrange the function making x the subject.

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To find the domain of (1/(x+1) + 3), identify the values of (x) that make the expression undefined. The expression is undefined when the denominator, (x + 1), is 0. Solve (x + 1 = 0), yielding (x = -1). Thus, the domain is all real numbers except (x = -1), which can be written as ((-∞, -1) \cup (-1, ∞)).

The range of (1/(x+1) + 3) can be found by considering the behavior of the function. As (x) approaches (-1) from the left or right, the term (1/(x+1)) approaches (-∞) or (+∞), respectively, causing the whole expression to trend towards (-∞) or (+∞). However, there is no (x)-value that will make (1/(x+1) + 3 = 0), due to the vertical asymptote at (x = -1) and the horizontal shift by 3 units upwards. To find the value that the range cannot take, set (1/(x+1) + 3 = 0), leading to (1/(x+1) = -3), and solve for (x), which doesn't lead to a restriction in the range directly but indicates the effect of the translation. The critical insight here is recognizing the function's behavior due to the asymptote and the vertical shift. Given this, it's clear the function covers all real numbers as its output, because for any value of (y), there's a corresponding (x) that will satisfy the equation (y = 1/(x+1) + 3). Therefore, the range is all real numbers, or ((-∞, +∞)). However, this general analysis might suggest reconsidering the specific transformation effects on the range, which typically for (1/(x + a) + k) forms, doesn't inherently exclude any real numbers unless specified by additional context or constraints.

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When evaluating a one-sided 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 one-sided 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 one-sided 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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