Determine the domain of the function f (x) = 1/3x^2+10x+8

The function is

f (x) = (1/3) x² + 10x + 8

Write the function in standard form for a parabola.

f (x) = (1/3) [x² + 30x] + 8

= (1/3) [ (x+15) ² - 225] + 8

= (1/3) (x+15) ² - 75 + 8

f (x) = (1/3) (x+15) ² - 67

This is a parabola with vertex at (-15, - 67).

The axis of symmetry is x = - 15

The curve opens upward because the coefficient of x² is positive.

As x - > - ∞, f - > + ∞.

As x - > + ∞, f - > + ∞

The domain is all real values of x (see the graph below).

Answer: The domain is (-∞, ∞)