Domain and Range of Function

Question

Hi! I don't know the domain and range of this function.

y=(x³+6x²-9x-54)/(x²+3x-10)

Answer 1

The domain is valid x values, the range is y values

In a rational function as here, domain will be all x, except where denominator is zero.

(x² + 3x - 10) can be factorised as (x+5) (x-2) so will be zero when x= -5 or x = 2

so domain is all real x, except x = -5 and x = 2

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For the range, you could plot the function on Desmos and spot it. Dividing numerator and denominator by x² gives x + 6 - 9/x² - 54/x³ / 1 + 3/x - 10/x²

as x > + infinity, y > + infinity

as x < - infinity. y < - infinity so from this looks like the range is all real numbers, but check on Desmos