If g(x)=x-2 is the inverse of the function f(x)=x+2, then graph of g(x) is the image of graph of f(x) about the line y=kx. Here k =

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to determine the value of \( k \) such that the graph of \( g(x) = x - 2 \) is the image of the graph of \( f(x) = x + 2 \) about the line \( y = kx \).

### Step-by-Step Solution:

1. **Identify the Functions**:
We have two functions:
- \( f(x) = x + 2 \)
- \( g(x) = x - 2 \)

2. **Graph the Functions**:
- For \( f(x) \):
- When \( x = 0 \), \( f(0) = 0 + 2 = 2 \) (point: \( (0, 2) \))
- When \( y = 0 \), \( 0 = x + 2 \) gives \( x = -2 \) (point: \( (-2, 0) \))
- The line passes through points \( (0, 2) \) and \( (-2, 0) \).

- For \( g(x) \):
- When \( x = 0 \), \( g(0) = 0 - 2 = -2 \) (point: \( (0, -2) \))
- When \( y = 0 \), \( 0 = x - 2 \) gives \( x = 2 \) (point: \( (2, 0) \))
- The line passes through points \( (0, -2) \) and \( (2, 0) \).

3. **Determine the Relationship Between the Two Functions**:
- The graph of \( g(x) \) is a downward shift of the graph of \( f(x) \) by 4 units. This can be seen as:
\[ g(x) = f(x) - 4 \]

4. **Find the Midpoint**:
- The midpoint between the two functions can be calculated as:
\[ \text{Midpoint} = \frac{f(x) + g(x)}{2} = \frac{(x + 2) + (x - 2)}{2} = \frac{2x}{2} = x \]
- This indicates that the line \( y = x \) is the line of symmetry.

5. **Determine the Value of \( k \)**:
- The line of symmetry \( y = x \) can be expressed in the form \( y = kx \), where \( k = 1 \).

### Final Answer:
Thus, the value of \( k \) is \( 1 \).