If `f:RtoR` is defined by `f(x)=ax+b,ane0` then `f^(-1)(x)`

Options

A. is given by `(x-b)/(a)`
B. is given by `(1)/(ax+b)`
C. does not exist as f is not onto
D. does not exist as f is not one-one

Answer

To find the inverse of the function \( f(x) = ax + b \) where \( a \neq 0 \), we can follow these steps:

### Step 1: Set the function equal to \( y \)
We start by letting:
\[ y = f(x) = ax + b \]

### Step 2: Solve for \( x \) in terms of \( y \)
Now, we need to isolate \( x \). First, we can rearrange the equation:
\[ ax = y - b \]
Next, we divide both sides by \( a \):
\[ x = \frac{y - b}{a} \]

### Step 3: Write the inverse function
Since we have expressed \( x \) in terms of \( y \), we can denote the inverse function \( f^{-1}(y) \):
\[ f^{-1}(y) = \frac{y - b}{a} \]

### Step 4: Replace \( y \) with \( x \)
To express the inverse function in terms of \( x \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{x - b}{a} \]

### Final Answer
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{x - b}{a} \]

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