Question

for the following function find (a) f(3), (b) $f\left(-\frac{1}{2}\
ight)$, (c) f(a), (d) $f\left(\frac{2}{m}\
ight)$, and (e) any values of x such that f(x) = 1.
$f(x)=\

$$\begin{cases}\\frac{2x + 2}{x - 3}&\\text{if }x\ eq3\\\\2&\\text{if }x = 3\\end{cases}$$

$
(type an integer or a simplified fraction.)
(b) find the value of $f\left(-\frac{1}{2}\
ight)$.
$f\left(-\frac{1}{2}\
ight)=-\frac{2}{7}$
(type an integer or a simplified fraction.)
(c) find the value of f(a).
$f(a)=\

$$\begin{cases}\\square&\\text{if }a\ eq3\\\\\\square&\\text{if }a = 3\\end{cases}$$

$

Explanation:

Step1: Calculate f(3)

Since \(x=3\), use \(f(x)=2\). Result: \(2\)

Step2: Calculate \(f(-\frac{1}{2})\)

Since \(-\frac{1}{2}≠3\), use \(f(x)=\frac{2x+2}{x-3}\). Substitute \(x=-\frac{1}{2}\):
\(\frac{2(-\frac{1}{2})+2}{-\frac{1}{2}-3}=\frac{-1+2}{-\frac{7}{2}}=\frac{1}{-\frac{7}{2}}=-\frac{2}{7}\)

Step3: Find f(a)

If \(a≠3\), use \(\frac{2a+2}{a-3}\); if \(a=3\), use \(2\).

Step4: Calculate \(f(\frac{2}{m})\)

Assume \(\frac{2}{m}≠3\) (unless \(m=\frac{2}{3}\)), use \(\frac{2(\frac{2}{m})+2}{\frac{2}{m}-3}=\frac{\frac{4}{m}+2}{\frac{2-3m}{m}}=\frac{4+2m}{2-3m}\)

Step5: Solve f(x)=1

Case1: \(x≠3\): \(\frac{2x+2}{x-3}=1\) → \(2x+2=x-3\) → \(x=-5\)
Case2: \(x=3\): \(f(3)=2≠1\). So \(x=-5\)

Answer:

(a) \(2\)
(b) \(-\frac{2}{7}\)
(c) \(f(a)=

$$\begin{cases}\frac{2a+2}{a-3} & \text{if }a≠3 \\ 2 & \text{if }a=3\end{cases}$$

\)
(d) \(\frac{2m+4}{2-3m}\) (for \(m≠\frac{2}{3}\))
(e) \(-5\)