If `f(x)=ax+b`, which of the following make(s) `f(x)=f^(-1)(x)`?
I. a=-1, b=any real number
II. A=1, b=0
III. A=any real number, b=0

Options

A. Only I
B. Only II
C. only III
D. Only I and II

Explanation

To solve the problem, we need to determine the conditions under which the function \( f(x) = ax + b \) is equal to its inverse \( f^{-1}(x) \).

### Step 1: Find the inverse of the function \( f(x) \)

We start with the equation:
\[ y = ax + b \]
To find the inverse, we will solve for \( x \) in terms of \( y \):
\[ y - b = ax \]
\[ x = \frac{y - b}{a} \]
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{x - b}{a} \]

### Step 2: Set \( f(x) \) equal to \( f^{-1}(x) \)

Now we set \( f(x) \) equal to \( f^{-1}(x) \):
\[ ax + b = \frac{x - b}{a} \]

### Step 3: Clear the fraction by multiplying through by \( a \)

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 the terms involving \( x \):
\[ a^2x - x + ab + b = 0 \]
Factoring out \( x \):
\[ (a^2 - 1)x + (ab + b) = 0 \]

### Step 5: Set the coefficients to zero

For the equation to hold for all \( x \), both coefficients must be zero:
1. \( a^2 - 1 = 0 \)
2. \( ab + b = 0 \)

### Step 6: Solve the first equation \( a^2 - 1 = 0 \)

From the first equation:
\[ a^2 = 1 \implies a = 1 \text{ or } a = -1 \]

### Step 7: Solve the second equation \( ab + b = 0 \)

Factoring out \( b \):
\[ b(a + 1) = 0 \]
This gives us two cases:
1. \( b = 0 \)
2. \( a + 1 = 0 \implies a = -1 \)

### Step 8: Analyze the cases

1. **If \( a = 1 \)**:
- From \( b(a + 1) = 0 \), since \( a + 1 = 2 \), we have \( b = 0 \).
- Thus, \( (a, b) = (1, 0) \).

2. **If \( a = -1 \)**:
- From \( b(a + 1) = 0 \), since \( a + 1 = 0 \), \( b \) can be any real number.
- Thus, \( (a, b) = (-1, b) \) where \( b \) is any real number.

### Conclusion

The valid pairs \((a, b)\) that satisfy \( f(x) = f^{-1}(x) \) are:
- \( (1, 0) \)
- \( (-1, b) \) where \( b \) is any real number.

### Answer to the options:
- I. \( a = -1, b = \text{any real number} \) (True)
- II. \( a = 1, b = 0 \) (True)
- III. \( a = \text{any real number}, b = 0 \) (False, since \( a \) cannot be any real number)

Thus, the correct options are I and II.

### Final Answer:
The correct options are I and II.