Question

If `u(x) = -2x^2+3 and v(x) = 1/x`, what is the range of`(u * v)(x)`?

Answer 1

`u@v (x) = u(v(x)) = u(1/x) = (3x^2-2)/x^2`

`"Range: " (-oo,3)`

Explanation:

`u(x) = -2x^2+3`

`v(x) = 1/x`

Whichever notation you prefer:
`(u@v) (x) = u(v(x))`

`u@v (x) = u(v(x)) = u(1/x)`

`u(1/x) = (-2)/x^2 + 3`

`u(1/x) = frac{3x^2-2}{x^2}`

The horizontal asymptote of this function is at `y=3`. So, the range of this function is from `(-oo,3)`