Find the value of `f ( f ( − 3 ) ) , if f ( x ) = (x+1)/ x`

Options

A. `7/3`
B. `5/3`
C. `5/2`
D. `3/2`

Answer

To find the value of \( f(f(-3)) \) given the function \( f(x) = \frac{x+1}{x} \), we will follow these steps:

### Step 1: Calculate \( f(-3) \)
We start by substituting \(-3\) into the function \( f(x) \).

\[ f(-3) = \frac{-3 + 1}{-3} \]

Calculating the numerator:

\[ -3 + 1 = -2 \]

Now substituting back into the function:

\[ f(-3) = \frac{-2}{-3} = \frac{2}{3} \]

### Step 2: Calculate \( f(f(-3)) \)
Now that we have \( f(-3) = \frac{2}{3} \), we need to find \( f\left(\frac{2}{3}\right) \).

Substituting \(\frac{2}{3}\) into the function \( f(x) \):

\[ f\left(\frac{2}{3}\right) = \frac{\frac{2}{3} + 1}{\frac{2}{3}} \]

Calculating the numerator:

\[ \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} \]

Now substituting back into the function:

\[ f\left(\frac{2}{3}\right) = \frac{\frac{5}{3}}{\frac{2}{3}} = \frac{5}{3} \times \frac{3}{2} = \frac{5}{2} \]

### Final Result
Thus, the value of \( f(f(-3)) \) is:

\[ \boxed{\frac{5}{2}} \]

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