If function `f:RtoR` is defined by `f(x)=3x-4` then `f^(-1)(x)` is given by

Options

A. `3x+4`
B. `(x)/(3)-4`
C. `(x+4)/(3)`
D. none of these

Answer

To find the inverse of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 3x - 4 \), we will follow these steps:

### Step 1: Check if the function is one-to-one (1-1)
A function is one-to-one if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).

Let’s assume:
\[ f(x_1) = f(x_2) \]
This implies:
\[ 3x_1 - 4 = 3x_2 - 4 \]
Now, we can simplify this equation:
\[ 3x_1 = 3x_2 \]
Dividing both sides by 3 gives:
\[ x_1 = x_2 \]
Since we have shown that \( f(x_1) = f(x_2) \) leads to \( x_1 = x_2 \), the function is indeed one-to-one.

### Step 2: Set \( f(x) = y \) and solve for \( x \)
Now, we will express \( f(x) \) in terms of \( y \):
\[ y = 3x - 4 \]
To find \( x \) in terms of \( y \), we rearrange the equation:
\[ 3x = y + 4 \]
Now, divide both sides by 3:
\[ x = \frac{y + 4}{3} \]

### Step 3: Write the inverse function
The expression we found for \( x \) in terms of \( y \) is the inverse function, which we denote as \( f^{-1}(y) \):
\[ f^{-1}(y) = \frac{y + 4}{3} \]
Since we typically write the inverse function in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{x + 4}{3} \]

### Conclusion
Thus, the inverse function \( f^{-1}(x) \) is given by:
\[ f^{-1}(x) = \frac{x + 4}{3} \]

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