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# Domain

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

Aug 16, 2023

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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
+121
0

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