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Find the domain of the function \[f(x)=\sqrt{-6x^2+11x+4} + \sqrt{x}.\]

 Aug 16, 2023

Best Answer 

 #1
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For a given function, the domain consists of all the values of \(x\) for which the function is defined. In this case, we need to consider two factors: the domain of the square root functions and the domain of the entire expression.

1. The square root functions are defined only for non-negative values under the square root. In other words, the expression inside the square root must be greater than or equal to 0. For the first square root:
   \[-6x^2 + 11x + 4 \geq 0.\]

   To solve this quadratic inequality, you can find the critical points by setting the expression equal to 0 and solving for \(x\):

   \[-6x^2 + 11x + 4 = 0.\]

   This factors as \((-3x + 4)(2x + 1) = 0\), giving solutions \(x = -1/2\) and \(x = 4/3\).

   Now, plot these critical points on a number line and choose test points to determine the sign of the expression within the intervals. You'll find that \(-6x^2 + 11x + 4 \geq 0\) when \(x \leq -1/2\) or \(x \geq 4/3\).

   For the second square root:
   \(x \geq 0\).

2. Combining both conditions from the square roots, the function \(f(x)\) is defined when:
   \[-6x^2 + 11x + 4 \geq 0 \quad \text{and} \quad x \geq 0.\]

   This means the domain of the function \(f(x)\) is the intersection of these intervals, which is \(x \geq 4/3\).

So, the domain of the function \(f(x)\) is \(\boxed{x \geq \frac{4}{3}}\).

 Aug 16, 2023
 #1
avatar+121 
0
Best Answer

For a given function, the domain consists of all the values of \(x\) for which the function is defined. In this case, we need to consider two factors: the domain of the square root functions and the domain of the entire expression.

1. The square root functions are defined only for non-negative values under the square root. In other words, the expression inside the square root must be greater than or equal to 0. For the first square root:
   \[-6x^2 + 11x + 4 \geq 0.\]

   To solve this quadratic inequality, you can find the critical points by setting the expression equal to 0 and solving for \(x\):

   \[-6x^2 + 11x + 4 = 0.\]

   This factors as \((-3x + 4)(2x + 1) = 0\), giving solutions \(x = -1/2\) and \(x = 4/3\).

   Now, plot these critical points on a number line and choose test points to determine the sign of the expression within the intervals. You'll find that \(-6x^2 + 11x + 4 \geq 0\) when \(x \leq -1/2\) or \(x \geq 4/3\).

   For the second square root:
   \(x \geq 0\).

2. Combining both conditions from the square roots, the function \(f(x)\) is defined when:
   \[-6x^2 + 11x + 4 \geq 0 \quad \text{and} \quad x \geq 0.\]

   This means the domain of the function \(f(x)\) is the intersection of these intervals, which is \(x \geq 4/3\).

So, the domain of the function \(f(x)\) is \(\boxed{x \geq \frac{4}{3}}\).

SpectraSynth Aug 16, 2023

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