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2 years ago

11

Chose all the values of x that are not in the domain of this rational function. Picture attached, 15 points and I'll give Brainl

iest! Due in two hours. A. -2 B. 1 C. 2 D. o

2 answers:

pogonyaev2 years ago

6 0

Answer:

All of them.

Step-by-step explanation:

For rational functions, the domain is all real numbers <em>except</em> for the zeros of the denominator.

Therefore, to find the x-values that are not in the domain, we need to solve for the zeros of the denominator. Therefore, set the denominator to zero:

$x(x-1)(x^2-4)=0$

Zero Product Property:

$x\neq 0\text{ or }x-1\neq 0\text{ or }x^2-4\neq 0$

Solve for the x in each of the three equations. The first one is already solved. Thus:

$x-1\neq 0 \text{ or }x^2-4\neq 0\\x\neq 1\text{ or }x^2\neq 4\\x\neq 1 \text{ or }x\neq\pm 2$

Thus, the values that <em>cannot</em> be in the domain of the rational function is:

$x=-2,0,1,2$

Click all the options.

zzz [600]2 years ago

3 0

Answer:

$\Large \boxed{\mathrm{All \ options}}$

Step-by-step explanation:

The domain of a function are all possible values for x.

To find the domain of a rational function, we set the denominator equal to 0, and solve for x. Those values of x are <em>not</em> included in the domain, since the denominator of 0 would make the rational function undefined.

$x(x-1)(x^2-4)=0$

Setting the factors equal to 0.

$x=0 \\ \\ \\\\ x-1=0 \\ \\ x=1 \\ \\ \\ \\ x^2 -4=0 \\ \\ x^2 =4 \\ \\ x=\pm 2$

The values of x that are <em>not</em> in the domain of the rational function are x=-2, x=1, x=2, and x=0.

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Answer:

3 solutions:

$\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}}$

Step-by-step explanation:

So, first of all, we need to figure the angles that cannot be included in our answers out. The only function in the equation that isn't defined for some angles is $tan(\frac{\theta}{2})$ so let's focus on that part of the equation first.

We know that:

$tan(\frac{\theta}{2})=\frac{sin(\frac{\theta}{2})}{cos(\frac{\theta}{2})}$

therefore:

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$cos(\frac{\theta}{2})=0$

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$\frac{\theta}{2}=\frac{\pi}{2}+\pi n$

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$\theta=\pi+2\pi n$

we can now start plugging values in for n:

$\theta=\pi+2\pi (0)=\pi$

if we plugged any value greater than 0, we would end up with an angle that is greater than $2\pi$ so, that's the only angle we cannot include in our answer set, so:

$\theta\neq \pi$

having said this, we can now start solving the equation:

$tan(\frac{\theta}{2})=sin(\theta)$

we can start solving this equation by using the half angle formula, such a formula tells us the following:

$tan(\frac{\theta}{2})=\frac{1-cos(\theta)}{sin(\theta)}$

so we can substitute it into our equation:

$\frac{1-cos(\theta)}{sin(\theta)}=sin(\theta)$

we can now multiply both sides of the equation by $sin(\theta)$

so we get:

$1-cos(\theta)=sin^{2}(\theta)$

we can use the pythagorean identity to rewrite $sin^{2}(\theta)$ in terms of cos:

$sin^{2}(\theta)=1-cos^{2}(\theta)$

so we get:

$1-cos(\theta)=1-cos^{2}(\theta)$

we can subtract a 1 from both sides of the equation so we end up with:

$-cos(\theta)=-cos^{2}(\theta)$

and we can now add $cos^{2}(\theta)$

to both sides of the equation so we get:

$cos^{2}(\theta)-cos(\theta)=0$

and we can solve this equation by factoring. We can factor $cos(\theta)$ to get:

$cos(\theta)(cos(\theta)-1)=0$

and we can use the zero product property to solve this, so we get two equations:

Equation 1:

$cos(\theta)=0$

$\theta=cos^{-1}(0)$

$\theta={\frac{\pi}{2}, \frac{3\pi}{2}}$

Equation 2:

$cos(\theta)-1=0$

we add a 1 to both sides of the equation so we get:

$cos(\theta)=1$

$\theta=cos^{-1}(1)$

$\theta=0$

so we end up with three answers to this equation:

$\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}}$

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