Determine the domain of the function? f(x) = sqrt(−x2+13x−40)

Question

I tried solving this one (I'm trying to only ask questions on here that i've tried solving lol), however, I keep getting the domain of (-infinity, infinity). However, my teacher said that that answer was incorrect. And help would be amazing!

Answer 1

The number under the square-root cannot be negative.  Therefore,

 

-x² + 13x - 40 ≥ 0

 

 

Divide both sides of the inequality by -1 to make the coefficient of x² positive 1.

 

x² - 13x + 40 ≤ 0

 

 

Here is the trick to solving this.  Replace the greater than or equal to sign with an equal sign.  This will give you the x values on the x-axis.

 

x² - 13x + 40 = 0

 

 

Factor using FOIL.

 

(x - 8)(x - 5) = 0

 

x = 5          and           x = 8

 

 

These x values lie on the x-axis.  But since the f(x) values need to be below the x-axis, the x values where this happens is in the interval

 

[5,8]

 

 

You can check this using a graphing calculator.