If `f: R to R ` defined as `f(x)=3x+7`, then find `f^(-1)(-2)`

Answer

To find \( f^{-1}(-2) \) for the function \( f(x) = 3x + 7 \), we will follow these steps:

### Step 1: Express the function in terms of \( y \)
Let \( f(x) = y \). Therefore, we have:
\[ y = 3x + 7 \]

### Step 2: Solve for \( x \) in terms of \( y \)
Rearranging the equation to solve for \( x \):
\[ 3x = y - 7 \]
\[ x = \frac{y - 7}{3} \]

### Step 3: Write the inverse function
From the equation above, we can express the inverse function:
\[ f^{-1}(y) = \frac{y - 7}{3} \]

### Step 4: Substitute \( y = -2 \) into the inverse function
Now, we need to find \( f^{-1}(-2) \):
\[ f^{-1}(-2) = \frac{-2 - 7}{3} \]

### Step 5: Simplify the expression
Calculating the numerator:
\[ -2 - 7 = -9 \]
Now, substituting back:
\[ f^{-1}(-2) = \frac{-9}{3} = -3 \]

### Conclusion
Thus, the value of \( f^{-1}(-2) \) is:
\[ \boxed{-3} \]