The functions f(x)=5(2)^(x) and g(x)=5(b)^(x) are graphed in the xy-plane. If the graph of function g is always increasing and f(x) g(x) for all x 0, then which of the following could be the value of b ? Choose 1 answer: (A) 0.25 (B) 1.25 (C) 2 (D) 5

Question

The functions  f(x)=5(2)^(x) and  g(x)=5(b)^(x) are graphed in the  xy-plane. If the graph of function  g is always increasing and  f(x) > g(x) for all  x > 0, then which of the following could be the value of  b ? Choose 1 answer: (A) 0.25 (B) 1.25 (C) 2 (D) 5

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Analyze Functions: Analyze the given functions and the conditions. We have two functions f(x) = 5(2)^x and g(x) = 5(b)^x. We are told that f(x) > g(x) for all x > 0, and the graph of g is always increasing. This means that b must be greater than 1 because if b were less than or equal to 1, the graph of g would not always be increasing.Compare Exponents: Compare the base of the exponents in f(x) and g(x). Since f(x) = 5(2)^x is greater than g(x) = 5(b)^x for all x > 0, the base of the exponent in f(x), which is 2, must be greater than the base of the exponent in g(x), which is b. Therefore, b must be less than 2.Determine Possible Values: Determine the possible values of b based on the given options. From the options given, we need to find a value of b that is greater than 1 (to ensure the graph of g is always increasing) and less than 2 (to ensure that f(x) > g(x) for all x > 0). Let's evaluate the options: (A) 0.25 - This would not result in an increasing graph for g(x). (B) 1.25 - This is greater than 1 and less than 2, so it could be a possible value for b. (C) 2 - This is equal to the base of the exponent in f(x), so f(x) would not be greater than g(x) for all x > 0. (D) 5 - This is greater than the base of the exponent in f(x), so f(x) would not be greater than g(x) for all x > 0.Choose Correct Answer: Choose the correct answer based on the analysis. The only value that satisfies the conditions that b is greater than 1 and less than 2 is option (B) 1.25.

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