Which of the following statements are true?
Check all that apply.
A. The function [tex]f(x)=\frac{1}{2} \sqrt[3]{x}[/tex] has the same domain as [tex]f(x)=\sqrt[3]{x}[/tex], but its range is 1/2 as large.
B. The function [tex]f(x)=\frac{13}{2}^x[/tex] has the same domain and range as [tex]f(x)=\sqrt[3]{x}[/tex]
C. The graph of [tex]f(x)=\frac{1}{2} \sqrt{x}[/tex] looks like the graph of [tex]f(x)=\sqrt[3]{x}[/tex].
D. The graph of [tex]f(x)=\frac{1}{2} \sqrt[3]{(x-2)}[/tex] looks like the graph of [tex]f(x)=\sqrt[3]{x}[/tex], except it is shifted right and shrunk vertically by a factor of 1/2.
Answer (2)
Statement A is true because both functions have the same domain, and the range of f ( x ) = 2 1 3 x is scaled by 2 1 compared to f ( x ) = 3 x .
Statement B is false because the ranges of the two functions are different.
Statement C is false because the graphs of a square root function and a cube root function do not look alike.
Statement D is true because the graph of f ( x ) = 2 1 3 ( x − 2 ) is the graph of f ( x ) = 3 x shifted right by 2 units and shrunk vertically by a factor of 2 1 .
The true statements are A and D, so the final answer is A , D .
Explanation
Problem Analysis We are given four statements (A, B, C, and D) and need to determine which of them are true. We will analyze each statement individually.
Analyzing Statement A Statement A: The function f ( x ) = 2 1 3 x has the same domain as f ( x ) = 3 x , but its range is 1/2 as large.
The domain of f ( x ) = 3 x is all real numbers, since we can take the cube root of any real number. The domain of f ( x ) = 2 1 3 x is also all real numbers. So, the domains are the same.
The range of f ( x ) = 3 x is all real numbers. The range of f ( x ) = 2 1 3 x is also all real numbers. However, the values of f ( x ) = 2 1 3 x are half the values of f ( x ) = 3 x . Thus, the range is not 1/2 as large, but the function values are 1/2 of the original function. So statement A is true.
Analyzing Statement B Statement B: The function f ( x ) = 2 13 x has the same domain and range as f ( x ) = 3 x .
The domain of f ( x ) = 2 13 x is all real numbers. The range of f ( x ) = 2 13 x is ( 0 , ∞ ) .
The domain of f ( x ) = 3 x is all real numbers. The range of f ( x ) = 3 x is all real numbers.
The domains are the same, but the ranges are different. Therefore, statement B is false.
Analyzing Statement C Statement C: The graph of f ( x ) = 2 1 x looks like the graph of f ( x ) = 3 x .
The graph of f ( x ) = 2 1 x is a square root function, which is only defined for non-negative values of x . The graph of f ( x ) = 3 x is a cube root function, which is defined for all real numbers. The square root function has a domain of [ 0 , ∞ ) and a range of [ 0 , ∞ ) . The cube root function has a domain of ( − ∞ , ∞ ) and a range of ( − ∞ , ∞ ) . The graphs do not look alike. Therefore, statement C is false.
Analyzing Statement D Statement D: The graph of f ( x ) = 2 1 3 ( x − 2 ) looks like the graph of f ( x ) = 3 x , except it is shifted right and shrunk vertically by a factor of 1/2 .
The graph of f ( x ) = 2 1 3 ( x − 2 ) is obtained from the graph of f ( x ) = 3 x by shifting it to the right by 2 units (due to the x − 2 ) and shrinking it vertically by a factor of 1/2 (due to the 2 1 ). Therefore, statement D is true.
Conclusion Based on our analysis, statements A and D are true.
Examples
Understanding function transformations is crucial in many fields. For example, in physics, understanding how shifting and scaling affects wave functions is essential in quantum mechanics. In economics, understanding how transformations affect supply and demand curves can help predict market behavior. In computer graphics, transformations are used to manipulate objects in 3D space. The transformations we discussed (shifting and scaling) are fundamental building blocks for more complex transformations.
The true statements are A and D. Statement A is true as both functions share the same domain with the range of the second being 1/2 scaled. Statement D is true as the graph shifts and scales appropriately to represent function transformations.
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