What is the domain and range of #sqrt((5x+6)/2)#?

Answer 1

Answer:

Domain #x in[-6/5,oo)#
Range #[0,oo)#

You must keep in mind that for the domain:

#sqrt(y)->y>=0#
#ln(y)->y>0#
#1/y->y!=0#

After that, you will be lead to an unequality giving you the domain.

This function is a combination of linear and square functions. Linear has domain #RR#. The square function though must have a positive number inside the square. Therefore:
#(5x+6)/2>=0#

Since 2 is positive:

#5x+6>=0#
#5x>= -6#

Since 5 is positive:

#x>= -6/5#

The domain of the functions is:

#x in[-6/5,oo)#
The range of the root function (outer function) is #[0,oo)# (infinite part can be proven through the limit as #x->oo#).
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Answer 2

The domain of the function ( f(x) = \sqrt{\frac{5x+6}{2}} ) is all real numbers such that ( \frac{5x+6}{2} \geq 0 ), which implies ( 5x + 6 \geq 0 ), so the domain is ( x \geq -\frac{6}{5} ) or in interval notation, ( \left[-\frac{6}{5}, \infty\right) ).

The range of ( f(x) ) is all real numbers greater than or equal to ( 0 ), because the square root of a non-negative number is always non-negative. So, the range is ( [0, \infty) ).

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Answer from HIX Tutor

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.

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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