Question
Which graph represents the function h(x)=−x√+2?
Responses
Responses
Answers
GPT-4o mini
It seems there might be a small error in the function you've provided, as "−x√" is not a standard notation. If you meant \( h(x) = -x\sqrt{x} + 2 \) (which combines \( -x \) and the square root of \( x \)), that would make sense as a function.
To accurately represent this function graphically, you can follow these steps:
1. **Identify the domain**: The function includes \( \sqrt{x} \), so \( x \) must be greater than or equal to 0 (i.e., \( x \geq 0 \)).
2. **Calculate intercepts**:
- **y-intercept**: When \( x = 0 \):
\[
h(0) = -0\sqrt{0} + 2 = 2
\]
- **x-intercept**: Set \( h(x) = 0 \) and solve for \( x \):
\[
-x\sqrt{x} + 2 = 0 \implies x\sqrt{x} = 2 \implies x^{3/2} = 2
\]
\[
x = \left(2\right)^{\frac{2}{3}} \approx 1.5874
\]
3. **Behavior of the function**: As \( x \) increases, \( -x\sqrt{x} \) will decrease rapidly and dominate the function due to the negative sign, which will eventually lead \( h(x) \) to decrease below zero after reaching a maximum point.
4. **Graph characteristics**: The graph will start at \( (0, 2) \), have a maximum somewhere between \( 0 < x < 2 \), and then it will decrease toward negative infinity as \( x \) increases further.
If you need to choose from multiple graph options, look for a graph that starts at 2 for \( x = 0 \), increases to a max point, and then decreases. If you can provide more context or the specific graphs to compare, I can help you identify the correct one.
To accurately represent this function graphically, you can follow these steps:
1. **Identify the domain**: The function includes \( \sqrt{x} \), so \( x \) must be greater than or equal to 0 (i.e., \( x \geq 0 \)).
2. **Calculate intercepts**:
- **y-intercept**: When \( x = 0 \):
\[
h(0) = -0\sqrt{0} + 2 = 2
\]
- **x-intercept**: Set \( h(x) = 0 \) and solve for \( x \):
\[
-x\sqrt{x} + 2 = 0 \implies x\sqrt{x} = 2 \implies x^{3/2} = 2
\]
\[
x = \left(2\right)^{\frac{2}{3}} \approx 1.5874
\]
3. **Behavior of the function**: As \( x \) increases, \( -x\sqrt{x} \) will decrease rapidly and dominate the function due to the negative sign, which will eventually lead \( h(x) \) to decrease below zero after reaching a maximum point.
4. **Graph characteristics**: The graph will start at \( (0, 2) \), have a maximum somewhere between \( 0 < x < 2 \), and then it will decrease toward negative infinity as \( x \) increases further.
If you need to choose from multiple graph options, look for a graph that starts at 2 for \( x = 0 \), increases to a max point, and then decreases. If you can provide more context or the specific graphs to compare, I can help you identify the correct one.