If f(x) `= sin theta. x + a` and the equatio f(x) `= f^(-1)(x)` is satisfied by every real value of x, then

Options

A. `theta = (pi)/(2)`
B. `theta = (3pi)/(2)`
C. `a in R`
D. `a = 1, theta = (pi)/(2)`

Answer

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

### Step 1: Find the Inverse Function \( f^{-1}(x) \)

1. Start with the equation for \( f(x) \):
\[ f(x) = \sin(\theta) \cdot x + a \]
2. To find the inverse, we set \( y = f(x) \):
\[ y = \sin(\theta) \cdot x + a \]
3. Rearranging for \( x \):
\[ y - a = \sin(\theta) \cdot x \]
\[ x = \frac{y - a}{\sin(\theta)} \]
4. Therefore, the inverse function is:
\[ f^{-1}(x) = \frac{x - a}{\sin(\theta)} \]

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

1. We set the two functions equal to each other:
\[ \sin(\theta) \cdot x + a = \frac{x - a}{\sin(\theta)} \]

### Step 3: Cross Multiply to Eliminate the Fraction

1. Cross multiplying gives:
\[ (\sin(\theta) \cdot x + a) \cdot \sin(\theta) = x - a \]
\[ \sin^2(\theta) \cdot x + a \cdot \sin(\theta) = x - a \]

### Step 4: Rearrange the Equation

1. Rearranging the equation:
\[ \sin^2(\theta) \cdot x - x + a \cdot \sin(\theta) + a = 0 \]
\[ x(\sin^2(\theta) - 1) + a(\sin(\theta) + 1) = 0 \]

### Step 5: Factor the Equation

1. We can factor the equation:
\[ x(\sin^2(\theta) - 1) + a(\sin(\theta) + 1) = 0 \]
This implies that for the equation to hold for all \( x \):
- The coefficient of \( x \) must be zero:
\[ \sin^2(\theta) - 1 = 0 \]
- The constant term must also be zero:
\[ a(\sin(\theta) + 1) = 0 \]

### Step 6: Solve for \( \theta \)

1. From \( \sin^2(\theta) - 1 = 0 \):
\[ \sin^2(\theta) = 1 \implies \sin(\theta) = \pm 1 \]
- This gives \( \theta = \frac{3\pi}{2} + 2k\pi \) or \( \theta = \frac{\pi}{2} + 2k\pi \) for any integer \( k \).

### Step 7: Solve for \( a \)

1. From \( a(\sin(\theta) + 1) = 0 \):
- If \( \sin(\theta) = 1 \) (which corresponds to \( \theta = \frac{\pi}{2} \)), then \( a \) can be any real number.
- If \( \sin(\theta) = -1 \) (which corresponds to \( \theta = \frac{3\pi}{2} \)), then \( a \) can also be any real number.

### Conclusion

Thus, the values we find are:
- \( \theta = \frac{3\pi}{2} \)
- \( a \) can be any real number \( a \in \mathbb{R} \)