Question

What are the domain and range of f (x) = -5/6 (x + 2)^2 - 8?

Answer 1

Option B

Domain: all real numbers

Range
`f(x) \le -8`

Explanation:

Given function is:

`f\left(x\right)=-(5)/(6)\:\left(x\:+\:2\right)^2\:-\:8`

The domain is the set of all x values that result in a real and defined value for f(x).

The function has no undefined points and x is free to range from -∞ to +∞

So the domain is -∞ < x <∞ which is
All real numbers

The range is the set of all possible values for f(x) given a specific domain. In order to determine the range without too much hard work, let's first examine the function f(x)

The above function is the equation for a parabola.

Some things to note about a parabola equation:

• The vertex form equation of a parabola is:
  
  `f(x) = a(x - h)^2 + k`
  where the vertex is the point (h, k). So the x-value of the vertex = h and corresponding y-value is
• A vertex is the maximum or minimum point in the parabola

• If the coefficient a > 0 then the parabola opens upward and the vertex i(h, k) s a minimum
• If the coefficient, a < 0 the parabola opens downward and the vertex (h, k) is a maximum
  

Let us compare the general vertex form equation of the parabola with the specific f(x) equation given

General Equation:
`a (x - h)^2 + k`

This example:
`-(5)/(6)\:\left(x\:+\:2\right)^2\:-\:8`

Comparing the similarity between the two equations we easily see that

`\boxed{a = -(5)/(6)}`
This means the parabola opens downward and (h, k) represents a max point in the parabola

`x - h = x + 2\\\\-h = 2\\\\h = -2\\`

`k = -8`

So the vertex is at (-2, -8)

Since -8 is the maximum value for f(x), and the parabola extends to infinity on both sides, the range is :

`\boxed{f(x) \le -8}`

The attached graph may help you understand better