Let's solve the given problem step by step for each part.
Given the function:
[tex]\[
q(x) = \frac{1}{x^2 - 9}
\][/tex]
### Part (a) [tex]\( q(0) \)[/tex]
To find [tex]\( q(0) \)[/tex]:
[tex]\[
q(0) = \frac{1}{0^2 - 9} = \frac{1}{-9} = -\frac{1}{9}
\][/tex]
We have:
[tex]\[
q(0) = -\frac{1}{9}
\][/tex]
So, the function value at [tex]\( q(0) \)[/tex] is [tex]\(-\frac{1}{9}\)[/tex].
### Part (b) [tex]\( q(3) \)[/tex]
To find [tex]\( q(3) \)[/tex]:
[tex]\[
q(3) = \frac{1}{3^2 - 9} = \frac{1}{9 - 9} = \frac{1}{0}
\][/tex]
Since division by zero is undefined:
[tex]\[
q(3) = \text{UNDEFINED}
\][/tex]
So, the function value at [tex]\( q(3) \)[/tex] is UNDEFINED.
### Part (c) [tex]\( q(y+3) \)[/tex]
To find [tex]\( q(y+3) \)[/tex]:
[tex]\[
q(y+3) = \frac{1}{(y+3)^2 - 9}
\][/tex]
First, simplify the denominator:
[tex]\[
(y+3)^2 - 9 = y^2 + 6y + 9 - 9 = y^2 + 6y
\][/tex]
Now, evaluate [tex]\( q(y+3) \)[/tex] and check specific values, like when [tex]\( y = 3 \)[/tex]:
[tex]\[
\text{For } y + 3 = 6:
q(6) = \frac{1}{6^2 - 9} = \frac{1}{36 - 9} = \frac{1}{27} = \frac{1}{27}
\][/tex]
Thus:
[tex]\[
q(y+3) = 0.037037037037037035 \text{ (numerically accurate value)}
\][/tex]
Therefore, the function value is:
[tex]\[
q(y+3) = 0.037037037037037035
\][/tex]
