What is the domain and range of #y =sqrt((x^2-5x-14))#?

1 Answer
Andy Y.
Feb 5, 2017

Domain: All #x<=-2# and #x>=7#
Range: All #y>=0#

Explanation:

The domain can be described as all the "legal" values of #x#.

  • You can't divide by zero
  • You can't have negatives under a square root

If you find the "illegal" values, then you know the domain is all #x# except those!

The "illegal" values of #x# would be whenever the mantissa #< 0#

#x^2-5x-14<0# ...illegal values are negatives under roots
#(x+2)(x-7)<0# ...factor the left hand side

Now separate the two factors and flip one of the inequalities. One of the terms has to be negative (i.e., #<0#) and the other must be positive (i.e., #>0#).

#x+2 > 0# and #x-7 < 0#
#x > -2# and #x < 7#

The domain is all #x# except those illegal ones you just found.

Domain: All #x<=-2# and all #x>7#

The range are all values of #y# with domain #x#'s plugged in.

Range: All #y>=0#

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