Which of the following correspondences can be called a function ?

Options

A. `f(x) = x^3 , {-1, 0, 1} to {0,1,2,3}`
B. `f(x)= pm sqrt x, {0,1,2}to{-2,-1,0,1,2}`
C. `f(x) = sqrtx, {0,1,4} to {-2,-1,0,1,2}`
D. `f(x) = -sqrtx, {0,1,4} to {-2,-1,0,1,2}`

Answer

To determine which of the given correspondences can be classified as a function, we need to apply the definition of a function. A correspondence \( f \) from a set \( X \) to a set \( Y \) is called a function if every element \( x \) in \( X \) has a unique image \( f(x) \) in \( Y \). This means that:

1. Each element in \( X \) must map to one and only one element in \( Y \).
2. It is acceptable for different elements in \( X \) to map to the same element in \( Y \), but no element in \( X \) can map to multiple elements in \( Y \).

Let's analyze each option step by step.

### Step 1: Analyze the first option \( f(x) = x^3 \)

- For \( x = -1 \): \( f(-1) = (-1)^3 = -1 \)
- For \( x = 0 \): \( f(0) = 0^3 = 0 \)
- For \( x = 1 \): \( f(1) = 1^3 = 1 \)

**Conclusion**: Each \( x \) maps to a unique \( f(x) \). Thus, this correspondence is a function.

### Step 2: Analyze the second option \( f(x) = \pm \sqrt{x} \)

- For \( x = 0 \): \( f(0) = \pm \sqrt{0} = 0 \) (maps to 0)
- For \( x = 1 \): \( f(1) = \pm \sqrt{1} = \pm 1 \) (maps to both 1 and -1)
- For \( x = 2 \): \( f(2) = \pm \sqrt{2} \) (maps to both \( \sqrt{2} \) and \( -\sqrt{2} \))

**Conclusion**: Since \( x = 1 \) maps to two different values (1 and -1), this correspondence is not a function.

### Step 3: Analyze the third option \( f(x) = \sqrt{x} \)

- For \( x = 0 \): \( f(0) = \sqrt{0} = 0 \)
- For \( x = 1 \): \( f(1) = \sqrt{1} = 1 \)
- For \( x = 4 \): \( f(4) = \sqrt{4} = 2 \)

**Conclusion**: Each \( x \) maps to a unique \( f(x) \). Thus, this correspondence is a function.

### Step 4: Analyze the fourth option \( f(x) = -\sqrt{x} \)

- For \( x = 0 \): \( f(0) = -\sqrt{0} = 0 \)
- For \( x = 1 \): \( f(1) = -\sqrt{1} = -1 \)
- For \( x = 4 \): \( f(4) = -\sqrt{4} = -2 \)

**Conclusion**: Each \( x \) maps to a unique \( f(x) \). Thus, this correspondence is a function.

### Final Summary of Results:
- Option 1: Function
- Option 2: Not a function
- Option 3: Function
- Option 4: Function