Answer :
Let's analyze each of the four given statements regarding the functions [tex]\( f(x)=2 \sqrt{x} \)[/tex], [tex]\( f(x)=-2 \sqrt{x} \)[/tex], [tex]\( f(x)=-\sqrt{x} \)[/tex], and [tex]\( f(x)=\frac{1}{2} \sqrt{x} \)[/tex] in relation to the function [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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
- Range: The range of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex].
For [tex]\( f(x)=2 \sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range is [tex]\( y \geq 0 \)[/tex], but the values are twice those of [tex]\( \sqrt{x} \)[/tex]. The smallest value is still 0, and it can become arbitrarily large as [tex]\( x \)[/tex] increases.
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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
For [tex]\( f(x)=-2 \sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range of [tex]\( -2 \sqrt{x} \)[/tex] is [tex]\( y \leq 0 \)[/tex] because multiplying by -2 will flip the positive values to the corresponding negative values. Thus, the range is entirely negative 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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
For [tex]\( f(x)=-\sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range is [tex]\( y \leq 0 \)[/tex] because multiplying by -1 flips the positive values to the corresponding negative values.
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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
For [tex]\( f(x)=\frac{1}{2} \sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range is [tex]\( y \geq 0 \)[/tex], but the values are half those of [tex]\( \sqrt{x} \)[/tex]. The smallest value is still 0, and it can become arbitrarily large as [tex]\( x \)[/tex] increases, though at half the rate.
Therefore, this statement is true.
Final Answer:
- [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 as [tex]\( f(x)=\sqrt{x} \)[/tex], but a different range.
- [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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
- Range: The range of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex].
For [tex]\( f(x)=2 \sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range is [tex]\( y \geq 0 \)[/tex], but the values are twice those of [tex]\( \sqrt{x} \)[/tex]. The smallest value is still 0, and it can become arbitrarily large as [tex]\( x \)[/tex] increases.
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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
For [tex]\( f(x)=-2 \sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range of [tex]\( -2 \sqrt{x} \)[/tex] is [tex]\( y \leq 0 \)[/tex] because multiplying by -2 will flip the positive values to the corresponding negative values. Thus, the range is entirely negative 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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
For [tex]\( f(x)=-\sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range is [tex]\( y \leq 0 \)[/tex] because multiplying by -1 flips the positive values to the corresponding negative values.
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: The domain of [tex]\( f(x)=\sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
For [tex]\( f(x)=\frac{1}{2} \sqrt{x} \)[/tex]:
- Domain: The domain is still [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative [tex]\( x \)[/tex].
- Range: The range is [tex]\( y \geq 0 \)[/tex], but the values are half those of [tex]\( \sqrt{x} \)[/tex]. The smallest value is still 0, and it can become arbitrarily large as [tex]\( x \)[/tex] increases, though at half the rate.
Therefore, this statement is true.
Final Answer:
- [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 as [tex]\( f(x)=\sqrt{x} \)[/tex], but a different range.
- [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.
