Given that `f (x)` is a non-constant linear function. Then the curves :

Options

A. `y = f(x) and y= f ^(-1)(x)` are orthogonal
B. `y = f (x) and y = f^(-1)(-x)` are orthogonal
C. `y = f (-x) and y = f^(-1)(x)` are orthogonal
D. `y=f (-x) and y= f^(-1)(-x)` are orthogonal

Answer

To solve the problem, we need to analyze the given information about the function \( f(x) \) and its inverse \( f^{-1}(x) \). We know that \( f(x) \) is a non-constant linear function, which can be expressed in the form:

\[ f(x) = mx + c \]

where \( m \neq 0 \) (since it is non-constant) and \( c \) is a constant.

### Step 1: Find the inverse function \( f^{-1}(x) \)

To find the inverse of the function, we start with the equation:

\[ y = f(x) = mx + c \]

Now, we solve for \( x \):

\[ y = mx + c \implies mx = y - c \implies x = \frac{y - c}{m} \]

Thus, the inverse function is:

\[ f^{-1}(x) = \frac{x - c}{m} \]

### Step 2: Determine the slopes of \( f(x) \) and \( f^{-1}(x) \)

The slope of \( f(x) \) is:

\[ m_1 = m \]

The slope of \( f^{-1}(x) \) can be found by differentiating \( f^{-1}(x) \):

\[ f^{-1}(x) = \frac{x - c}{m} \]

Thus, the slope \( m_2 \) is:

\[ m_2 = \frac{1}{m} \]

### Step 3: Check for orthogonality

Two lines are orthogonal (perpendicular) if the product of their slopes is -1:

\[ m_1 \cdot m_2 = m \cdot \frac{1}{m} = 1 \]

Since \( 1 \neq -1 \), \( f(x) \) and \( f^{-1}(x) \) are not orthogonal.

### Step 4: Analyze the other options

1. **Option B**: \( y = f(x) \) and \( y = f^{-1}(-x) \)

Substitute \( -x \) into the inverse function:

\[ f^{-1}(-x) = \frac{-x - c}{m} \]

The slope of \( f^{-1}(-x) \) is still \( \frac{1}{m} \), but we need to check the slopes:

The slope of \( f(x) \) remains \( m \), and the slope of \( f^{-1}(-x) \) becomes:

\[ m_2 = \frac{1}{m} \]

The product of the slopes:

\[ m \cdot \left(-\frac{1}{m}\right) = -1 \]

Thus, they are orthogonal.

2. **Option C**: \( y = f(-x) \) and \( y = f^{-1}(x) \)

Substitute \( -x \) into \( f(x) \):

\[ f(-x) = m(-x) + c = -mx + c \]

The slope of \( f(-x) \) is \( -m \) and the slope of \( f^{-1}(x) \) is \( \frac{1}{m} \).

The product of the slopes:

\[ (-m) \cdot \left(\frac{1}{m}\right) = -1 \]

Thus, they are also orthogonal.

### Conclusion

From the analysis, we find that options B and C are correct as both pairs of functions are orthogonal.