# How do you find the domain and range of #y= 2 |x+1|-1#?

The domain is all real numbers.

The range is

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To find the domain and range of the function ( y = 2 |x + 1| - 1 ), we first determine the domain by considering all possible values of ( x ) that make the function defined. Since the absolute value function ( |x + 1| ) is defined for all real numbers, there are no restrictions on the domain of ( x ).

Therefore, the domain of the function is all real numbers.

Next, to find the range, we analyze the behavior of the absolute value function ( |x + 1| ). The absolute value of any real number is always non-negative. Therefore, ( |x + 1| \geq 0 ) for all values of ( x ).

Since ( y = 2 |x + 1| - 1 ) involves multiplying the absolute value function by 2 and then subtracting 1, the smallest possible value of ( y ) occurs when ( |x + 1| = 0 ), which results in ( y = -1 ).

As ( |x + 1| ) increases, ( y ) increases at twice the rate, but it is always reduced by 1. Therefore, the range of the function is all real numbers greater than or equal to ( -1 ).

Therefore, the range of the function is ( y \geq -1 ).

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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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