If `f(x)` is a function `f:R-> R`, we say `f(x)` has property I. If `f(f(x)) =x` for all real numbers x. II. `f(-f(x))=-x` for all real numbers x. How many linear functions, have both property I and Il ?

Options

A. 0
B. 2
C. 3
D. Infinite

Answer

To solve the problem, we need to find the linear functions \( f(x) \) that satisfy the two given properties:

1. \( f(f(x)) = x \) for all real numbers \( x \).
2. \( f(-f(x)) = -x \) for all real numbers \( x \).

### Step 1: Assume the form of the linear function

Let’s assume that \( f(x) \) is a linear function of the form:
\[ f(x) = ax + b \]
where \( a \) and \( b \) are constants.

### Step 2: Apply the first property

Now, we apply the first property \( f(f(x)) = x \):
\[ f(f(x)) = f(ax + b) = a(ax + b) + b = a^2x + ab + b \]
Setting this equal to \( x \):
\[ a^2x + ab + b = x \]
This gives us two equations by comparing coefficients:
1. \( a^2 = 1 \)
2. \( ab + b = 0 \)

### Step 3: Solve the first set of equations

From the first equation \( a^2 = 1 \), we find:
\[ a = 1 \quad \text{or} \quad a = -1 \]

Now, we will analyze each case for \( a \).

### Case 1: \( a = 1 \)

Substituting \( a = 1 \) into the second equation:
\[ 1 \cdot b + b = 0 \implies 2b = 0 \implies b = 0 \]
Thus, one possible function is:
\[ f(x) = 1 \cdot x + 0 = x \]

### Case 2: \( a = -1 \)

Substituting \( a = -1 \) into the second equation:
\[ -1 \cdot b + b = 0 \implies -b + b = 0 \implies 0 = 0 \]
This does not provide any restriction on \( b \), so \( b \) can be any real number. Thus, another possible function is:
\[ f(x) = -1 \cdot x + b = -x + b \]

### Step 4: Apply the second property

Now we apply the second property \( f(-f(x)) = -x \):
1. For \( f(x) = x \):
\[ f(-f(x)) = f(-x) = -x \quad \text{(This holds true)} \]

2. For \( f(x) = -x + b \):
\[ f(-f(x)) = f(-(-x + b)) = f(x - b) = -(x - b) + b = -x + b + b = -x + 2b \]
Setting this equal to \( -x \):
\[ -x + 2b = -x \implies 2b = 0 \implies b = 0 \]
Thus, this function simplifies to:
\[ f(x) = -x \]

### Conclusion

The two linear functions that satisfy both properties are:
1. \( f(x) = x \)
2. \( f(x) = -x \)

Thus, there are **2 linear functions** that have both properties.

### Final Answer
The number of linear functions that have both properties I and II is **2**.

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