If `f(x)=(2x)/(5)-3` for all `x in R` then: `f^(-1)(-5)=`

Options

A. 7
B. -11
C. -5
D. none of these.

Answer

To solve the problem, we need to find the inverse of the function \( f(x) = \frac{2x}{5} - 3 \) and then evaluate it at \( -5 \).

### Step-by-Step Solution:

1. **Set up the equation**:
We start with the function:
\[ y = f(x) = \frac{2x}{5} - 3 \]

2. **Isolate \( x \)**:
To find the inverse function, we need to express \( x \) in terms of \( y \). First, we add 3 to both sides:
\[ y + 3 = \frac{2x}{5} \]

3. **Multiply by 5**:
Next, we multiply both sides by 5 to eliminate the fraction:
\[ 5(y + 3) = 2x \]

4. **Solve for \( x \)**:
Now, we divide both sides by 2:
\[ x = \frac{5(y + 3)}{2} \]

5. **Express the inverse function**:
Thus, we can write the inverse function as:
\[ f^{-1}(y) = \frac{5(y + 3)}{2} \]

6. **Evaluate \( f^{-1}(-5) \)**:
Now we need to find \( f^{-1}(-5) \):
\[ f^{-1}(-5) = \frac{5(-5 + 3)}{2} \]

7. **Simplify the expression**:
Calculate the expression inside the parentheses:
\[ -5 + 3 = -2 \]
Now substitute this back into the equation:
\[ f^{-1}(-5) = \frac{5(-2)}{2} \]

8. **Final calculation**:
Simplify:
\[ f^{-1}(-5) = \frac{-10}{2} = -5 \]

### Conclusion:
The value of \( f^{-1}(-5) \) is \( -5 \).