Question

What is the domain and range of `g(x) = x^2 + 7x -18`?

Answer 1

Domain is all `x in RR`
Range is `{yinRR|y>=-121/4}=[-121/4;oo)`

Explanation:

This is a 2nd degree quadratic polynomial so its graph is a parabola.

Its general form is `y=ax^2+bx+c` where in this case a = 1 indicating that the arms go up, b = 7, c = - 18 indicating the graph has y-intercept at - 18.

The domain is all possible x values that are allowed as inputs and so in this case is all real numbers `RR` .

The range is all possible output y values that are allowed and so since the turning point occurs when the derivative equals zero,
`=>2x+7=0=>x=-7/2`
The corresponding y value is then `g(-7/2)=-121/4`

Hence the range `{yinRR|y>=-121/4}=[-121/4;oo)`

I have included the graph underneath for extra clarity.

graph{x^2+7x-18 [-65.77, 65.9, -32.85, 32.9]}