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

y = (x³ + 6x² - 9x - 54) / (x² + 3x - 10)

Domain (set of all possible input values, i.e. all values of argument x is allowed to take).

x³+ 6x² - 9x - 54 x³ + 6x² - 9x - 54

y = ---------------------- = -----------------------

x² + 3x - 10 (x + 5)(x - 2)

Dividing by 0 is undefined, therefore x can't be -5 or 2, hence:

Domain of this function is all real numbers, except -5 and 2.

Using set notation, we can put it: {∀ x ∈ R | x ≠ -5, x ≠ 2}

Range (set of all possible output values, i.e. all values of function y produced by all the domain values):

To find the range we'll start with dividing the leading terms of the polynomials in numerator and denominator:

x³

---- = x, and note that coefficient is positive, which tells us that

x²

When x tends to infinity, y tends to infinity, x → ∞ , y → ∞

and

when x tends to negative infinity, y tends to negative infinity, x → -∞ , y → -∞

Range of this function is all real numbers.

Using set notation, we can put it: {∀ y ∈ R}

The answer can also be stated with the interval notation

Domain: (-∞, -5) ∪ (-5, 2) ∪ (2, ∞)

Range: (-∞, ∞)

Answer 2

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