Determine all real values of the variable for which the function has the value indicated.
-
B) -9, 3
C) -3, 9
D) no real numbers

Question

Determine all real values of the variable for which the function has the value indicated.
-
B) -9, 3 
C) -3, 9 
D) no real numbers

Quizplus · Accepted Answer

We can see that the given function is a quadratic function, so it will have real roots if its discriminant is non-negative.
$\sqrt{t+9} - \sqrt{t+3} = 2$
Squaring both sides we get,
$t+9 - 2\sqrt{(t+9)(t+3)} + t+3 = 4$
$\sqrt{(t+9)(t+3)} = t+4$
Squaring again and simplifying, we get
$t^2 - 23t - 60 \geq 0$
$(t-(-3))(t-9) \geq 0$
$t \in (-\infty, -3] \cup [9, \infty)$
This is the required range for the function. Now, we need to check if there exists any value of t for which the function takes the value 2.
Plugging in t = 3, we get
$\sqrt{3+9} - \sqrt{3+3} = 3-2 = 1$
Plugging in t = -9, we get
$\sqrt{-9+9} - \sqrt{-9+3} = 0 - \sqrt{-6}$
This is not a real number. So the function does not take the value of 2 for any value of t in the given domain. Thus, the answer is B.

Determine All Real Values of the Variable for Which the Function