If f(x)=ax+b and the equation `f(x)=f^(-1)(x)` be satisfied by every real value of x, then

Options

A. a=2, b=-1
B. `a=-1,b in R`
C. `a=1, b in R`
D. a=1, b=-1

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the equation \( f(x) = f^{-1}(x) \) holds for every real value of \( x \). Given that \( f(x) = ax + b \), we will follow these steps:

### Step 1: Find the Inverse Function
We start with the function:

\[ f(x) = ax + b \]

To find the inverse function \( f^{-1}(x) \), we set \( y = f(x) \):

\[ y = ax + b \]

Now, we solve for \( x \):

\[ y - b = ax \implies x = \frac{y - b}{a} \]

Thus, the inverse function is:

\[ f^{-1}(x) = \frac{x - b}{a} \]

### Step 2: Set the Functions Equal
We need to satisfy the condition \( f(x) = f^{-1}(x) \):

\[ ax + b = \frac{x - b}{a} \]

### Step 3: Clear the Fraction
To eliminate the fraction, we multiply both sides by \( a \):

\[ a(ax + b) = x - b \]

This simplifies to:

\[ a^2x + ab = x - b \]

### Step 4: Rearrange the Equation
Now, we rearrange the equation to group all terms involving \( x \) on one side:

\[ a^2x - x + ab + b = 0 \]

Factoring out \( x \):

\[ (a^2 - 1)x + (ab + b) = 0 \]

### Step 5: Set Coefficients to Zero
For this equation to hold for every real value of \( x \), both coefficients must be zero:

1. \( a^2 - 1 = 0 \)
2. \( ab + b = 0 \)

### Step 6: Solve the First Equation
From \( a^2 - 1 = 0 \):

\[ a^2 = 1 \implies a = 1 \text{ or } a = -1 \]

### Step 7: Solve the Second Equation
From \( ab + b = 0 \):

\[ b(a + 1) = 0 \]

This gives us two cases:

1. \( b = 0 \)
2. \( a + 1 = 0 \implies a = -1 \)

### Step 8: Analyze the Cases
- If \( a = 1 \), then \( b \) can be any real number.
- If \( a = -1 \), then \( b \) must be \( 0 \).

### Conclusion
Thus, the values of \( a \) and \( b \) that satisfy the original condition are:

- \( a = -1 \) and \( b \) can be any real number.
- Or \( a = 1 \) with \( b \) being any real number.

### Final Result
The solution can be summarized as:

- \( a = -1 \) and \( b \in \mathbb{R} \)