If `f(x)=(3x)/(7)+2` for all `x in R`, then: `f^-1(1)=`

Options

A. `-6`
B. `-7`
C. `-24`
D. none of these.

Answer

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

### Step 1: Set \( f(x) \) equal to \( y \)
Let \( f(x) = y \). Therefore, we have:
\[ y = \frac{3x}{7} + 2 \]

### Step 2: Rearrange the equation to solve for \( x \)
To find the inverse function, we need to express \( x \) in terms of \( y \). First, we can isolate the term involving \( x \):
\[ y - 2 = \frac{3x}{7} \]

### Step 3: Eliminate the fraction
Multiply both sides by 7 to eliminate the denominator:
\[ 7(y - 2) = 3x \]

### Step 4: Solve for \( x \)
Now, divide both sides by 3 to solve for \( x \):
\[ x = \frac{7(y - 2)}{3} \]

### Step 5: Substitute \( y = 1 \) to find \( f^{-1}(1) \)
Now we need to find \( f^{-1}(1) \):
\[ f^{-1}(1) = \frac{7(1 - 2)}{3} \]
\[ = \frac{7(-1)}{3} \]
\[ = -\frac{7}{3} \]

### Conclusion
Thus, the value of \( f^{-1}(1) \) is:
\[ f^{-1}(1) = -\frac{7}{3} \]

### Final Answer
Since \(-\frac{7}{3}\) does not match any of the provided options, the correct answer is "none of these".

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