Question

hw - properties of rational functions
question 2, check r6.1.5
determine the domain of the rational expression.
\\(\frac{z^{2}-4}{z^{2}+2z - 63}\\)
select the correct answer below and if necessary, fill in the answer box to complete your choice.
a. \\(\\{z|z\
eq - 7,5\\}\\) (use a comma to separate answers as needed.)
b. \\(\\{z|z\\) is any real number\\}

Explanation:

Step1: Recall domain rule for rational functions

The domain of a rational function $\frac{f(z)}{g(z)}$ is all real - numbers except where $g(z)=0$. Here, $g(z)=z^{2}+2z - 63$.

Step2: Set the denominator equal to zero

We solve the equation $z^{2}+2z - 63 = 0$.

Step3: Factor the quadratic equation

$z^{2}+2z - 63=(z + 9)(z - 7)=0$.

Step4: Solve for z

Using the zero - product property, if $(z + 9)(z - 7)=0$, then $z+9 = 0$ or $z - 7=0$. So $z=-9$ or $z = 7$.

Answer:

The domain is $\{z|z
eq - 9,7\}$. Since the given options are incorrect, the correct domain in set - builder notation is $\{z|z
eq - 9,7\}$. If we assume there was a mistake in the problem - solving process in the options and we re - check the factoring of $z^{2}+2z - 63=(z + 9)(z - 7)$ and the values that make the denominator zero are $z=-9$ and $z = 7$. If we assume the options were mis - typed and we consider the closest correct form among the given ones, the correct way to write the domain based on the options' format should be an option like $\{z|z
eq - 9,7\}$ but since it's not there, there is an error in the options provided. If we had to choose from the given ones, both A and B are incorrect as written.