# How do I find the range of the function #y=-2^x+2#?

Here's my attempt:

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To find the range of the function ( y = -2^x + 2 ), we first need to understand the behavior of the exponential function ( -2^x ).

The range of an exponential function ( a^x ), where ( a ) is a constant and ( x ) varies over all real numbers, depends on the value of ( a ):

- If ( 0 < a < 1 ), the range is ( (0, +\infty) ), meaning the function's values are positive and approach zero but never reach it.
- If ( a = 1 ), the range is ( {1} ), because ( 1^x = 1 ) for all real numbers ( x ).
- If ( a > 1 ), the range is ( (-\infty, +\infty) ), meaning the function's values cover all real numbers.

In this case, ( a = -2 ), so ( -2^x ) will have all negative values except when ( x ) is even, in which case it will be positive.

Adding 2 to ( -2^x ) shifts the graph vertically upward by 2 units.

Thus, the range of the function ( y = -2^x + 2 ) is all real numbers, except that it's shifted upward by 2 units, so it does not reach values less than 2. Therefore, the range is ( (-\infty, 2] ).

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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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