How does the range of [tex]g(x)=\frac{6}{x}[/tex] compare with the range of the parent function [tex]f(x)=\frac{1}{x}[/tex] ?
A. The range of both [tex]f(x)[/tex] and [tex]g(x)[/tex] is all real numbers
B. The range of both [tex]f(x)[/tex] and [tex]g(x)[/tex] is all nonzero real numbers
C. The range of [tex]f(x)[/tex] is all real numbers, the range of [tex]g(x)[/tex] is all real numbers except 6
D. The range of [tex]f(x)[/tex] is all nonzero real numbers, the range of [tex]g(x)[/tex] is all real numbers except 6
Answer (2)
Determine the range of the parent function f ( x ) = x 1 , which is all nonzero real numbers.
Determine the range of the transformed function g ( x ) = x 6 , which is also all nonzero real numbers.
Compare the two ranges and conclude that both functions have the same range.
The range of both f ( x ) and g ( x ) is all nonzero real numbers, so the answer is: The range of both f ( x ) and g ( x ) is all nonzero real numbers .
Explanation
Understanding the Problem We are asked to compare the ranges of the functions f ( x ) = f r a c 1 x and g ( x ) = f r a c 6 x . The range of a function is the set of all possible output values (y-values) that the function can produce. We need to determine the range of each function and then compare them to the given options.
Finding the Range of f(x) For the parent function f ( x ) = f r a c 1 x , we can see that x can take any real value except 0 , since division by zero is undefined. As x approaches infinity, f ( x ) approaches 0 , and as x approaches 0 , f ( x ) approaches infinity. Also, f ( x ) can take any value except 0 , since there is no value of x for which f r a c 1 x = 0 . Therefore, the range of f ( x ) is all real numbers except 0 .
Finding the Range of g(x) For the transformed function g ( x ) = f r a c 6 x , similar to f ( x ) , x can take any real value except 0 . As x approaches infinity, g ( x ) approaches 0 , and as x approaches 0 , g ( x ) approaches infinity. Also, g ( x ) can take any value except 0 , since there is no value of x for which f r a c 6 x = 0 . Therefore, the range of g ( x ) is all real numbers except 0 .
Comparing the Ranges Comparing the ranges of f ( x ) and g ( x ) , we see that both functions have the same range: all real numbers except 0 . Therefore, the correct statement is: The range of both f ( x ) and g ( x ) is all nonzero real numbers.
Examples
Understanding the ranges of functions like f ( x ) = f r a c 1 x and g ( x ) = f r a c 6 x is crucial in various real-world applications. For instance, in physics, these functions can model inverse relationships, such as the relationship between the intensity of light and the distance from the source. Knowing the range helps determine the possible values of intensity. In economics, they can represent supply and demand curves, where understanding the range helps analyze market behavior and predict price fluctuations. Moreover, in computer science, these functions can be used in algorithm analysis to understand the efficiency and limitations of certain algorithms.
The ranges of both functions f ( x ) = x 1 and g ( x ) = x 6 are all nonzero real numbers. Therefore, the chosen multiple-choice option is: B. The range of both functions is all nonzero real numbers.
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