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# Domain

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Okay, so I've never eally been clear on how to find domain and range of a function. Could someone explain how to get them then demonstrate on the problem \({4}^{2x}-5\)

Oct 13, 2017
edited by AdamTaurus  Oct 13, 2017

#1
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42x  - 5

This is an exponential.......the domain for an exponential is all real numbers  [ note....2x is defined for all real numbers....so.....42x  is, too....because the exponent is just a real number ]

The range is trickier.......if it was just    42x  the range would be  (0, infinity)

However.......the " - 5 "   part  shifts the graph down by 5 units.....so.....the range now becomes

(-5, infinity)

See the graph here :  https://www.desmos.com/calculator/szmhu8kwjm

Notice that graph almost touches   y = -5    [ but not quite  !!!]

Oct 13, 2017
#2
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Could you also possibly do the domain and range of -sqrt(2x-6).

#3
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y  = - √ [ 2x -6]

First.....since we can't take the square root of something < 0....then  2x  - 6  must be ≥ 0

So

2x  - 6 ≥  0

2x ≥ 6

x ≥ 3     and this is the domain  →    [3, infinity )

The range is a little tougher

The  greatest  value that  y can be is when x = 3......which means that y = 0

Note that all x's greater than 3 produce  a larger and larger square root....but...the " - " out front means that  we gett larger and larger negative square roots......so......the range is actually

( - infinity, 0 ]

See the graph here :  https://www.desmos.com/calculator/0mevks4btb

Note that as x increases, y decreases

Oct 13, 2017