Let f(x) be a linear function which maps [-1, 1] onto [0, 2], then f(x) can be

Options

A. `x+1`
B. `-x-1`
C. `-x+1`
D. `x-1`

Answer

To solve the problem, we need to find the linear function \( f(x) \) that maps the interval \([-1, 1]\) onto the interval \([0, 2]\). A linear function can be expressed in the form:

\[ f(x) = ax + b \]

where \( a \neq 0 \).

### Step 1: Determine the Function's Behavior
Since \( f(x) \) is a linear function, it can either be increasing or decreasing. We will consider both cases.

### Case 1: \( f(x) \) is an Increasing Function
1. **Minimum Input**: When \( x = -1 \), the minimum value of \( f(x) \) should be \( 0 \).
\[ f(-1) = a(-1) + b = -a + b = 0 \quad \text{(Equation 1)} \]

2. **Maximum Input**: When \( x = 1 \), the maximum value of \( f(x) \) should be \( 2 \).
\[ f(1) = a(1) + b = a + b = 2 \quad \text{(Equation 2)} \]

### Step 2: Solve the Equations
From Equation 1:
\[ b = a \]

Substituting \( b = a \) into Equation 2:
\[ a + a = 2 \implies 2a = 2 \implies a = 1 \]
Now substituting \( a = 1 \) back into \( b = a \):
\[ b = 1 \]

Thus, the function is:
\[ f(x) = 1x + 1 = x + 1 \]

### Case 2: \( f(x) \) is a Decreasing Function
1. **Minimum Input**: When \( x = -1 \), the maximum value of \( f(x) \) should be \( 2 \).
\[ f(-1) = a(-1) + b = -a + b = 2 \quad \text{(Equation 3)} \]

2. **Maximum Input**: When \( x = 1 \), the minimum value of \( f(x) \) should be \( 0 \).
\[ f(1) = a(1) + b = a + b = 0 \quad \text{(Equation 4)} \]

### Step 3: Solve the New Equations
From Equation 4:
\[ b = -a \]

Substituting \( b = -a \) into Equation 3:
\[ -a + (-a) = 2 \implies -2a = 2 \implies a = -1 \]
Now substituting \( a = -1 \) back into \( b = -a \):
\[ b = 1 \]

Thus, the function is:
\[ f(x) = -1x + 1 = -x + 1 \]

### Conclusion
The two possible linear functions that map \([-1, 1]\) onto \([0, 2]\) are:
1. \( f(x) = x + 1 \) (increasing function)
2. \( f(x) = -x + 1 \) (decreasing function)