Answer :
Sure, let's examine each statement step-by-step:
1. Statement: [tex]\(f(x)=2 \sqrt{x}\)[/tex] has the same domain and range as [tex]\(f(x)=\sqrt{x}\)[/tex].
- Domain:
Both functions [tex]\(f(x)=2 \sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(f(x)\)[/tex] also goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex]. Therefore, the range is [tex]\([0, \infty)\)[/tex].
- For [tex]\(f(x)=2 \sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(2\sqrt{x}\)[/tex] also goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex]. Therefore, the range is [tex]\([0, \infty)\)[/tex].
- The ranges and domains of both functions are the same.
- Therefore, this statement is true.
2. Statement: [tex]\(f(x)=-2 \sqrt{x}\)[/tex] has the same domain and range as [tex]\(f(x)=\sqrt{x}\)[/tex].
- Domain:
Both functions [tex]\(f(x)=-2\sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], the range is [tex]\([0, \infty)\)[/tex], as already determined.
- For [tex]\(f(x)=-2 \sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(-2\sqrt{x}\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(-\infty\)[/tex]. Thus, the range is [tex]\((-\infty, 0]\)[/tex].
- The ranges of both functions are different.
- Therefore, this statement is false.
3. Statement: [tex]\(f(x)=-\sqrt{x}\)[/tex] has the same domain as [tex]\(f(x)=\sqrt{x}\)[/tex], but a different range.
- Domain:
Both functions [tex]\(f(x)=-\sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], the range is [tex]\([0, \infty)\)[/tex].
- For [tex]\(f(x)=-\sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(-\sqrt{x}\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(-\infty\)[/tex]. Thus, its range is [tex]\((-\infty, 0]\)[/tex].
- The ranges of both functions are different.
- Therefore, this statement is true.
4. Statement: [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:
Both functions [tex]\(f(x)=\frac{1}{2} \sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], the range is [tex]\([0, \infty)\)[/tex].
- For [tex]\(f(x)=\frac{1}{2} \sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(\frac{1}{2}\sqrt{x}\)[/tex] also goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex]. Therefore, the range is [tex]\([0, \infty)\)[/tex].
- The ranges of both functions are the same.
- Therefore, this statement is false.
Based on this analysis, the corresponding true/false results for each statement are:
(1, 0, 1, 0)
1. Statement: [tex]\(f(x)=2 \sqrt{x}\)[/tex] has the same domain and range as [tex]\(f(x)=\sqrt{x}\)[/tex].
- Domain:
Both functions [tex]\(f(x)=2 \sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(f(x)\)[/tex] also goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex]. Therefore, the range is [tex]\([0, \infty)\)[/tex].
- For [tex]\(f(x)=2 \sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(2\sqrt{x}\)[/tex] also goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex]. Therefore, the range is [tex]\([0, \infty)\)[/tex].
- The ranges and domains of both functions are the same.
- Therefore, this statement is true.
2. Statement: [tex]\(f(x)=-2 \sqrt{x}\)[/tex] has the same domain and range as [tex]\(f(x)=\sqrt{x}\)[/tex].
- Domain:
Both functions [tex]\(f(x)=-2\sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], the range is [tex]\([0, \infty)\)[/tex], as already determined.
- For [tex]\(f(x)=-2 \sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(-2\sqrt{x}\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(-\infty\)[/tex]. Thus, the range is [tex]\((-\infty, 0]\)[/tex].
- The ranges of both functions are different.
- Therefore, this statement is false.
3. Statement: [tex]\(f(x)=-\sqrt{x}\)[/tex] has the same domain as [tex]\(f(x)=\sqrt{x}\)[/tex], but a different range.
- Domain:
Both functions [tex]\(f(x)=-\sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], the range is [tex]\([0, \infty)\)[/tex].
- For [tex]\(f(x)=-\sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(-\sqrt{x}\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(-\infty\)[/tex]. Thus, its range is [tex]\((-\infty, 0]\)[/tex].
- The ranges of both functions are different.
- Therefore, this statement is true.
4. Statement: [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:
Both functions [tex]\(f(x)=\frac{1}{2} \sqrt{x}\)[/tex] and [tex]\(f(x)=\sqrt{x}\)[/tex] are defined for [tex]\(x \geq 0\)[/tex], so their domains are the same: [tex]\([0, \infty)\)[/tex].
- Range:
- For [tex]\(f(x)=\sqrt{x}\)[/tex], the range is [tex]\([0, \infty)\)[/tex].
- For [tex]\(f(x)=\frac{1}{2} \sqrt{x}\)[/tex], as [tex]\(x\)[/tex] goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex], [tex]\(\frac{1}{2}\sqrt{x}\)[/tex] also goes from [tex]\(0\)[/tex] to [tex]\(\infty\)[/tex]. Therefore, the range is [tex]\([0, \infty)\)[/tex].
- The ranges of both functions are the same.
- Therefore, this statement is false.
Based on this analysis, the corresponding true/false results for each statement are:
(1, 0, 1, 0)
