Which of the following statements are true? Check all of the boxes that apply.
[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.
Answer (2)
Determine the domain and range of f ( x ) = x : Domain is [ 0 , ∞ ) , range is [ 0 , ∞ ) .
Analyze f ( x ) = 2 x : Same domain and range as f ( x ) = x .
Analyze f ( x ) = − x : Same domain as f ( x ) = x , but a different range ( − ∞ , 0 ] .
The true statements are: ' f ( x ) = 2 x has the same domain and range as f ( x ) = x ' and ' f ( x ) = − x has the same domain as f ( x ) = x , but a different range'. T r u e
Explanation
Analyzing the Base Function We need to analyze the domains and ranges of the given functions and compare them to f ( x ) = x . Let's start by finding the domain and range of the base function.
Domain and Range of f(x) = sqrt(x) For f ( x ) = x , the domain is all non-negative real numbers, since we can only take the square root of non-negative numbers. So, the domain is x ≥ 0 , or [ 0 , ∞ ) . The range is also all non-negative real numbers, since the square root of a non-negative number is always non-negative. So, the range is f ( x ) ≥ 0 , or [ 0 , ∞ ) .
Domain and Range of f(x) = 2sqrt(x) Now let's analyze f ( x ) = 2 x . The domain is still all non-negative real numbers, x ≥ 0 , or [ 0 , ∞ ) . The range is also all non-negative real numbers, but scaled by a factor of 2. So, the range is f ( x ) ≥ 0 , or [ 0 , ∞ ) . Comparing this to f ( x ) = x , we see that the domain and range are the same. Therefore, the statement ' f ( x ) = 2 x has the same domain and range as f ( x ) = x ' is true.
Domain and Range of f(x) = -2sqrt(x) Next, let's analyze f ( x ) = − 2 x . The domain is still all non-negative real numbers, x ≥ 0 , or [ 0 , ∞ ) . However, the range is now all non-positive real numbers, since we are multiplying the square root by -2. So, the range is f ( x ) ≤ 0 , or ( − ∞ , 0 ] . Comparing this to f ( x ) = x , we see that the domain is the same, but the range is different. Therefore, the statement ' f ( x ) = − 2 x has the same domain and range as f ( x ) = x ' is false.
Domain and Range of f(x) = -sqrt(x) Now, let's analyze f ( x ) = − x . The domain is still all non-negative real numbers, x ≥ 0 , or [ 0 , ∞ ) . The range is all non-positive real numbers, since we are taking the negative of the square root. So, the range is f ( x ) ≤ 0 , or ( − ∞ , 0 ] . Comparing this to f ( x ) = x , we see that the domain is the same, but the range is different. Therefore, the statement ' f ( x ) = − x has the same domain as f ( x ) = x , but a different range' is true.
Domain and Range of f(x) = 1/2 sqrt(x) Finally, let's analyze f ( x ) = 2 1 x . The domain is still all non-negative real numbers, x ≥ 0 , or [ 0 , ∞ ) . The range is also all non-negative real numbers, but scaled by a factor of 2 1 . So, the range is f ( x ) ≥ 0 , or [ 0 , ∞ ) . Comparing this to f ( x ) = x , we see that the domain is the same, but the range is different in magnitude (but not in the set of numbers). However, since both ranges are [ 0 , ∞ ) , they are considered the same. Therefore, the statement ' f ( x ) = 2 1 x has the same domain as f ( x ) = x , but a different range' is false.
Final Answer In conclusion, the true statements are:
f ( x ) = 2 x has the same domain and range as f ( x ) = x .
f ( x ) = − x has the same domain as f ( x ) = x , but a different range.
Examples
Understanding the domain and range of functions is crucial in many real-world applications. For example, when modeling the height of a projectile over time, the domain represents the time interval during which the projectile is in the air, and the range represents the possible heights it can reach. Similarly, in economics, the domain of a cost function might represent the number of units produced, and the range represents the total cost of production. By analyzing the domain and range, we can gain valuable insights into the behavior and limitations of the model.
The true statements are that f ( x ) = 2 x has the same domain and range as f ( x ) = x , and f ( x ) = − x has the same domain as f ( x ) = x , but a different range. The remaining statements are false as they do not match the domain and/or range comparisons. Therefore, the true checks are the first and third statements.
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