Question

Given f (x) = x2 + 2x – 6 and values of the linear function g(x) in the table, what is the range of (f + g)(x)?

Answer 1

First, let's use the values on the table to find the linear function f(x).

A linear function passing through points (a, g(a)) and (b, g(b)) using the following equation:

`g(x)-g(a)=(g(b)-g(a))/(b-a)(x-a)`

Since the line describing g(x) passes through points (-6, 14) and (-3, 8), we have:

a = -6

g(a) = 14

b = -3

g(b) = 8

So, g(x) is given by:

`\begin{gathered} g(x)-14=(8-14)/(-3-(-6))(x-(-6)) \\ \\ g(x)-14=(-6)/(-3+6)(x+6) \\ \\ g(x)-14=-2(x+6)_{} \\ \\ g(x)=-2x-12+14 \\ \\ g(x)=-2x+2 \end{gathered}`

Now, we need to compute (f+g)(x) to find its range.

We have:

`(f+g)(x)=f(x)+g(x)=x^(2)+2x-6-2x+2=x^(2)-4`

Since the roots of (f+g)(x) are -2 and 2, the vertice of the parabola has x-coordinate 0 (the middle between the roots). At x = 0, we have:

`(f+g)(x)=0^(2)-4=-4`

Also, since the coefficient of x² is 1, which is positive, this function represents a parabola opened upwards.

So, the minimum value of that function is y = -4. So, the range of the function is all real values equal to or greater than -4.

In interval notation, the range is written as

`(-4,\infty)`