Question
compare and contrast both functions
Y= square root of x and y = -square root of x + 1 + 5
Y= square root of x and y = -(√x + 1 + 5)
Y= square root of x and y = -square root of x + 1 + 5
Y= square root of x and y = -(√x + 1 + 5)
Answers
GPT-4o mini
Let's analyze and compare the two functions provided:
1. **Function Definitions**:
- **Function 1**: \( y = \sqrt{x} \)
- **Function 2**: \( y = -\sqrt{x} + 1 + 5 \)
- **Function 3**: \( y = -(\sqrt{x} + 1 + 5) \)
### Characteristics of Each Function
**Function 1: \( y = \sqrt{x} \)**
- Domain: \( x \geq 0 \) (since we only consider the square root of non-negative numbers)
- Range: \( y \geq 0 \) (output is always non-negative)
- Shape: This function represents a curve that starts at the origin (0, 0) and increases slowly and continuously towards the right, forming the right half of a parabola.
**Function 2: \( y = -\sqrt{x} + 1 + 5 \)**
- This can be simplified to \( y = -\sqrt{x} + 6 \).
- Domain: \( x \geq 0 \)
- Range: \( y \leq 6 \) (the highest point occurs when \( x = 0 \), giving \( y = 6 \), and decreases as \( x \) increases)
- Shape: This function is the reflection of \( y = \sqrt{x} \) across the x-axis, shifted up by 6 units. It starts at \( (0, 6) \) and decreases as \( x \) increases, creating a downward-opening curve.
**Function 3: \( y = -(\sqrt{x} + 1 + 5) \)**
- This can be simplified to \( y = -\sqrt{x} - 6 \).
- Domain: \( x \geq 0 \)
- Range: \( y \leq -6 \) (the lowest point occurs when \( x = 0 \), giving \( y = -6 \), and decreases as \( x \) increases)
- Shape: This function is also the reflection of \( y = \sqrt{x} \) across the x-axis, but shifted downward by 6 units. It starts at \( (0, -6) \) and decreases further as \( x \) increases.
### Comparison Summary
- **Domains**: All three functions have the same domain of \( x \geq 0 \).
- **Range**:
- \( y = \sqrt{x} \): \( y \geq 0 \)
- \( y = -\sqrt{x} + 6 \): \( y \leq 6 \)
- \( y = -(\sqrt{x} + 1 + 5) \): \( y \leq -6 \)
- **Shapes**:
- \( y = \sqrt{x} \) is an upward-opening curve starting at the origin.
- \( y = -\sqrt{x} + 6 \) is a downward-opening curve starting at \( (0, 6) \).
- \( y = -(\sqrt{x} + 1 + 5) \) is also a downward-opening curve starting at \( (0, -6) \).
### Conclusively
- Functions 2 and 3 are downward-opening transformations of the square root function, reflecting its basic shape.
- Function 1 is the original square root function, increasing positively.
- Functions 2 and 3 represent various shifts and reflections of the original function in different directions.
1. **Function Definitions**:
- **Function 1**: \( y = \sqrt{x} \)
- **Function 2**: \( y = -\sqrt{x} + 1 + 5 \)
- **Function 3**: \( y = -(\sqrt{x} + 1 + 5) \)
### Characteristics of Each Function
**Function 1: \( y = \sqrt{x} \)**
- Domain: \( x \geq 0 \) (since we only consider the square root of non-negative numbers)
- Range: \( y \geq 0 \) (output is always non-negative)
- Shape: This function represents a curve that starts at the origin (0, 0) and increases slowly and continuously towards the right, forming the right half of a parabola.
**Function 2: \( y = -\sqrt{x} + 1 + 5 \)**
- This can be simplified to \( y = -\sqrt{x} + 6 \).
- Domain: \( x \geq 0 \)
- Range: \( y \leq 6 \) (the highest point occurs when \( x = 0 \), giving \( y = 6 \), and decreases as \( x \) increases)
- Shape: This function is the reflection of \( y = \sqrt{x} \) across the x-axis, shifted up by 6 units. It starts at \( (0, 6) \) and decreases as \( x \) increases, creating a downward-opening curve.
**Function 3: \( y = -(\sqrt{x} + 1 + 5) \)**
- This can be simplified to \( y = -\sqrt{x} - 6 \).
- Domain: \( x \geq 0 \)
- Range: \( y \leq -6 \) (the lowest point occurs when \( x = 0 \), giving \( y = -6 \), and decreases as \( x \) increases)
- Shape: This function is also the reflection of \( y = \sqrt{x} \) across the x-axis, but shifted downward by 6 units. It starts at \( (0, -6) \) and decreases further as \( x \) increases.
### Comparison Summary
- **Domains**: All three functions have the same domain of \( x \geq 0 \).
- **Range**:
- \( y = \sqrt{x} \): \( y \geq 0 \)
- \( y = -\sqrt{x} + 6 \): \( y \leq 6 \)
- \( y = -(\sqrt{x} + 1 + 5) \): \( y \leq -6 \)
- **Shapes**:
- \( y = \sqrt{x} \) is an upward-opening curve starting at the origin.
- \( y = -\sqrt{x} + 6 \) is a downward-opening curve starting at \( (0, 6) \).
- \( y = -(\sqrt{x} + 1 + 5) \) is also a downward-opening curve starting at \( (0, -6) \).
### Conclusively
- Functions 2 and 3 are downward-opening transformations of the square root function, reflecting its basic shape.
- Function 1 is the original square root function, increasing positively.
- Functions 2 and 3 represent various shifts and reflections of the original function in different directions.