Questions: Given that f(x)=-3x-2/x-2 f(1)= f(-7)= f(1/2)=

Transcript text: Given that $\mathrm{f}(x)=\frac{-3 x-2}{x-2}$ \[ f(1)= \] $\square$ \[ f(-7)= \] $\square$ \[ f\left(\frac{1}{2}\right)= \]

Solution

Solution Steps

To find the values of $ f(x) $ for the given function $ f(x) = \frac{-3x - 2}{x - 2} $ at specific points, we need to substitute the given values of $ x $ into the function and simplify.

Substitute $ x = 1 $ into the function and simplify.
Substitute $ x = -7 $ into the function and simplify.
Substitute $ x = \frac{1}{2} $ into the function and simplify.

Step 1: Calculate $ f(1) $

To find $ f(1) $, we substitute $ x = 1 $ into the function: \[ f(1) = \frac{-3(1) - 2}{1 - 2} = \frac{-3 - 2}{-1} = \frac{-5}{-1} = 5.0 \]

Step 2: Calculate $ f(-7) $

Next, we calculate $ f(-7) $ by substituting $ x = -7 $: \[ f(-7) = \frac{-3(-7) - 2}{-7 - 2} = \frac{21 - 2}{-9} = \frac{19}{-9} \approx -2.1111 \]

Step 3: Calculate $ f\left(\frac{1}{2}\right) $

Finally, we find $ f\left(\frac{1}{2}\right) $ by substituting $ x = \frac{1}{2} $: \[ f\left(\frac{1}{2}\right) = \frac{-3\left(\frac{1}{2}\right) - 2}{\frac{1}{2} - 2} = \frac{-\frac{3}{2} - 2}{\frac{1}{2} - \frac{4}{2}} = \frac{-\frac{3}{2} - \frac{4}{2}}{-\frac{3}{2}} = \frac{-\frac{7}{2}}{-\frac{3}{2}} = \frac{7}{3} \approx 2.3333 \]