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Sagot :
Let's analyze each function and compare their domains and ranges with that of [tex]\( f(x) = \sqrt{x} \)[/tex].
1. [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex] because multiplying by 2 doesn't change the non-negative nature of the output values.
Therefore, this statement is true.
2. [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex] because multiplying by -2 flips the range to cover all non-positive values.
Therefore, this statement is false.
3. [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex].
Therefore, this statement is true.
4. [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = \frac{1}{2}\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex], but the values are scaled down by a factor of [tex]\(\frac{1}{2}\)[/tex].
Therefore, this statement is true.
Final Answer:
The statements that are true are:
- [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
So, the correct checkboxes are:
- ✓ [tex]\( f(x)=2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x)=\sqrt{x} \)[/tex].
- [tex]\( f(x)=-2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x)=\sqrt{x} \)[/tex].
- ✓ [tex]\( f(x)=-\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x)=\sqrt{x} \)[/tex], but a different range.
- ✓ [tex]\( f(x)=\frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x)=\sqrt{x} \)[/tex], but a different range.
1. [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex] because multiplying by 2 doesn't change the non-negative nature of the output values.
Therefore, this statement is true.
2. [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex] because multiplying by -2 flips the range to cover all non-positive values.
Therefore, this statement is false.
3. [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex].
Therefore, this statement is true.
4. [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = \frac{1}{2}\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex], but the values are scaled down by a factor of [tex]\(\frac{1}{2}\)[/tex].
Therefore, this statement is true.
Final Answer:
The statements that are true are:
- [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
So, the correct checkboxes are:
- ✓ [tex]\( f(x)=2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x)=\sqrt{x} \)[/tex].
- [tex]\( f(x)=-2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x)=\sqrt{x} \)[/tex].
- ✓ [tex]\( f(x)=-\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x)=\sqrt{x} \)[/tex], but a different range.
- ✓ [tex]\( f(x)=\frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x)=\sqrt{x} \)[/tex], but a different range.
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