El conjunto de positividad de f(x) = x² − 49x2 es

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5. The set of positivity of $f(x) = x^4 - 49x^2$ is where $f(x) > 0$ .

Factor the function.
$f(x) = x^2(x^2 - 49) = x^2(x-7)(x+7)$
Find the roots of $f(x)=0$ .
The roots are $x=0$ (with multiplicity 2), $x=7$ , and $x=-7$ .
Analyze the sign of $f(x)$ in the intervals defined by the roots.
Since $x^2 \ge 0$ for all real $x$ , the sign of $f(x)$ is determined by $(x-7)(x+7)$ , except at $x=0$ where $f(0)=0$ .
The expression $(x-7)(x+7)$ is positive when $x < -7$ or $x > 7$ .
Therefore, $f(x) > 0$ for $x \in (-\infty, -7) \cup (7, +\infty)$ .
The correct option is $\boxed{(-\infty;-7) \cup (7;+\infty)}$ .

6. The vertical asymptote of $f(x) = \frac{12x-6}{2x-8}$ occurs where the denominator is zero and the numerator is non-zero.
Set the denominator equal to zero.
$2x - 8 = 0$ $2x = 8$ $x = 4$
Check the numerator at $x=4$ .
$12(4) - 6 = 48 - 6 = 42$
Since the numerator is $42 \ne 0$ at $x=4$ , there is a vertical asymptote at $x=4$ .
The correct option is $\boxed{x=4}$ .

7. Given $f(x) = 5x+1$ and $g(x) = e^{1-x}$ , we need to find the value of $f \circ g(1)$ .
Calculate $g(1)$ .
$g(1) = e^{1-1} = e^0 = 1$
Calculate $f(g(1))$ , which is $f(1)$ .
$f(1) = 5(1) + 1 = 5 + 1 = 6$
The value of $f \circ g(1)$ is 6.
The correct option is $\boxed{6}$ .

8. We need to find all solutions of the equation $\sin(x) = -\frac{1}{2}$ in the interval $(0, 2\pi)$ .
Find the reference angle $\alpha$ such that $\sin(\alpha) = \frac{1}{2}$ .
$\alpha = \frac{\pi}{6}$
Determine the quadrants where $\sin(x)$ is negative.
Sine is negative in the third and fourth quadrants.
Find the solutions in the interval $(0, 2\pi)$ .
In the third quadrant, the solution is $x_1 = \pi + \alpha$ .
$x_1 = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}$
In the fourth quadrant, the solution is $x_2 = 2\pi - \alpha$ .
$x_2 = 2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6}$
The set of solutions is $\left\{\frac{7\pi}{6}, \frac{11\pi}{6}\right\}$ .
The correct option is $\boxed{\left\{\frac{7\pi}{6}, \frac{11\pi}{6}\right\}}$ .

El conjunto de positividad de f(x) = x² − 49x2 es

Step-by-step solution

Factor the function.

Find the roots of $f(x)=0$ .

Analyze the sign of $f(x)$ in the intervals defined by the roots.

Set the denominator equal to zero.

Check the numerator at $x=4$ .

Calculate $g(1)$ .

Calculate $f(g(1))$ , which is $f(1)$ .

Find the reference angle $\alpha$ such that $\sin(\alpha) = \frac{1}{2}$ .

Determine the quadrants where $\sin(x)$ is negative.

Find the solutions in the interval $(0, 2\pi)$ .

Still stuck on this one?

El conjunto de positividad de f(x) = x² − 49x2 es

Step-by-step solution

Factor the function.

Find the roots of f(x)=0f(x)=0f(x)=0.

Analyze the sign of f(x)f(x)f(x) in the intervals defined by the roots.

Set the denominator equal to zero.

Check the numerator at x=4x=4x=4.

Calculate g(1)g(1)g(1).

Calculate f(g(1))f(g(1))f(g(1)), which is f(1)f(1)f(1).

Find the reference angle α\alphaα such that sin⁡(α)=12\sin(\alpha) = \frac{1}{2}sin(α)=21​.

Determine the quadrants where sin⁡(x)\sin(x)sin(x) is negative.

Find the solutions in the interval (0,2π)(0, 2\pi)(0,2π).

Still stuck on this one?

Find the roots of $f(x)=0$ .

Analyze the sign of $f(x)$ in the intervals defined by the roots.

Check the numerator at $x=4$ .

Calculate $g(1)$ .

Calculate $f(g(1))$ , which is $f(1)$ .

Find the reference angle $\alpha$ such that $\sin(\alpha) = \frac{1}{2}$ .

Determine the quadrants where $\sin(x)$ is negative.

Find the solutions in the interval $(0, 2\pi)$ .