To solve the given questions for the function
[tex]\[ f(x) = \begin{cases}
\frac{2x+1}{x-8} & \text{if } x \neq 8 \\
6 & \text{if } x = 8
\end{cases}, \][/tex]
we will determine the following values:
### (a) Find the value of [tex]\( f(8) \)[/tex].
According to the definition of the function, when [tex]\( x = 8 \)[/tex], we use the second part of the piecewise function:
[tex]\[ f(8) = 6 \][/tex]
So the value of [tex]\( f(8) \)[/tex] is:
[tex]\[ f(8) = 6 \][/tex]
### (b) Find the value of [tex]\( f\left(-\frac{1}{2}\right) \)[/tex].
To find [tex]\( f\left(-\frac{1}{2}\right) \)[/tex], we use the first part of the piecewise function because [tex]\( x \neq 8 \)[/tex]:
[tex]\[ f\left(-\frac{1}{2}\right) = \frac{2\left(-\frac{1}{2}\right) + 1}{\left(-\frac{1}{2}\right) - 8} \][/tex]
Simplifying inside the fraction:
[tex]\[ f\left(-\frac{1}{2}\right) = \frac{-1 + 1}{-\frac{1}{2} - 8} = \frac{0}{-\frac{17}{2}} = 0 \][/tex]
So, the value of [tex]\( f\left(-\frac{1}{2}\right) \)[/tex] is:
[tex]\[ f\left(-\frac{1}{2}\right) = -0 \][/tex]
### (c) Find the value of [tex]\( f(a) \)[/tex].
For an arbitrary value [tex]\( a \)[/tex], as long as [tex]\( a \neq 8 \)[/tex], we use:
[tex]\[ f(a) = \frac{2a + 1}{a - 8} \][/tex]
### (d) Find the value of [tex]\( f\left(\frac{2}{m}\right) \)[/tex].
Again, for any value [tex]\( \frac{2}{m} \neq 8 \)[/tex]:
[tex]\[ f\left(\frac{2}{m}\right) = \frac{2\left(\frac{2}{m}\right) + 1}{\left(\frac{2}{m}\right) - 8} \][/tex]
Simplifying,
[tex]\[ f\left(\frac{2}{m}\right) = \frac{\frac{4}{m} + 1}{\frac{2}{m} - 8} \][/tex]
[tex]\[ f\left(\frac{2}{m}\right) = \frac{\frac{4 + m}{m}}{\frac{2 - 8m}{m}} = \frac{4 + m}{2 - 8m} \][/tex]
### (e) Possible values of [tex]\( x \)[/tex] such that [tex]\( f(x) \neq T(x) \)[/tex].
Given that [tex]\( f(x) \)[/tex] is defined differently when [tex]\( x = 8 \)[/tex] compared to when [tex]\( x \neq 8 \)[/tex], the values for [tex]\( x \)[/tex] such that [tex]\( f(x) \)[/tex] is defined are [tex]\( x \neq 8 \)[/tex].
Thus, evaluating [tex]\( f(x) \)[/tex] for a range of values [tex]\( x \)[/tex] (excluding [tex]\( x = 8 \)[/tex]) gives the sequence:
[tex]\[ [1.0556, 1.0, 0.9375, 0.8667, 0.7857, 0.6923, 0.5833, 0.4545, 0.3, 0.1111, -0.125, -0.4286, -0.8333, -1.4, -2.25, -3.6667, -6.5, -15.0, 19.0] \][/tex]
So the values of [tex]\( x \neq 8 \)[/tex] for which [tex]\( f(x) \)[/tex] is calculated can be observed in this list.
Summarizing all the results:
(a) [tex]\( f(8) = 6 \)[/tex]
(b) [tex]\( f\left( -\frac{1}{2} \right) = -0 \)[/tex]
(c) [tex]\( f(a) = \frac{2a + 1}{a - 8} \)[/tex]
(d) [tex]\( f\left( \frac{2}{m} \right) = \frac{4 + m}{2 - 8m} \)[/tex]
(e) Values of [tex]\( x \)[/tex] such that [tex]\( T(x) \)[/tex] are all [tex]\( x \neq 8 \)[/tex] in the given range and corresponding evaluated values.
