Question

What is the range of the function `f(x) = -2(6^x)+3`?

Answer 1

`(-oo, 3)`

Explanation:

The parent function: `g(x) = 6^x`
It has:
`y-"intercept": (0, 1)`
When `x-> -oo, y -> 0` so, there is a horizontal asymptote at `y = 0`, the `x`-axis.
When `x-> oo, y -> oo`.

For the function `f(x) = -2(6^x)`:
`y-"intercept": (0, -2)`
When `x-> -oo, y -> 0` so there is a horizontal asymptote at `y = 0`, the `x`-axis.
Because of the `-2` coefficient, the function turns downward:
When `x-> oo, y -> -oo`.

For the function `f(x) = -2(6^x) + 3`
`y-"intercept": (0, 1)`
When `x-> -oo, y -> 3` so there is a horizontal asymptote at `y = 3`.
Because of the `-2` coefficient, the function turns downward:
When `x-> oo, y -> -oo`.

Therefore the range (valid `y`-values): `(-oo, 3)`