Question

What is the domain, range, y intercepts and absolute maximum of `f(x)=8+2x-x^2`?

Answer 1

Domain `(-oo, +oo)`, Range `(-oo, +9)`, y-intercept `+8`, Maximum `+9`

Explanation:

`f(x) = 8+2x-x^2`

`f(x)` is a quadratic function, defined and continious `forall x in RR`

Hence, the domain of `f(x)` is `(-oo, +oo)`

The `y` intercept occurs at `x=0`

`-> y=8+0+0 = 8`

We know that the quadratic has an absloute maximum since the coefficient of `x^2 <0`

`f'(x) = 2-2x`

The absoulte maximum of `f(x)` occurs at `f'(x) = 0`

`-> 2x=2`

i.e `x=1`

`:. f(x)_"max" = 8+2-1 = 9`

Since `f(x)` has no finite minimum the range of `f(x)` is `(-oo, 9)`

These results can be realised by the graph of `f(x)` below:

graph{8+2x-x^2 [-15.97, 16.07, -4.59, 11.42]}