The graph of y=-g(x) in the xy -plane always has a negative slope and passes through the origin. If g is an exponential function, which of the following could define g ? Choose 1 answer: (A) g(x)=-(2)^(x)+1 (B) g(x)=-((3)/(4))^(x)+1 (C) g(x)=((2)/(5))^(x)-1 (D) g(x)=3^(x)

Question

The graph of  y=-g(x) in the  xy -plane always has a negative slope and passes through the origin. If  g is an exponential function, which of the following could define  g ? Choose 1 answer: (A)  g(x)=-(2)^(x)+1 (B)  g(x)=-((3)/(4))^(x)+1 (C)  g(x)=((2)/(5))^(x)-1 (D)  g(x)=3^(x)

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Find Exponential Function: We need to find a function g(x) such that when we take y = -g(x), the graph will have a negative slope and pass through the origin. Since g is an exponential function, it should be of the form g(x) = a^x, where a is a positive constant. The negative sign in front of g(x) will ensure that the slope of y = -g(x) is negative. To pass through the origin, the function must not have any vertical shifts, meaning there should be no constant term added or subtracted from a^x.Evaluate Option (A): Let's evaluate option (A) g(x) = -(2)^x + 1. If we take y = -g(x), we get y = -(-(2)^x + 1) = (2)^x - 1. This function does not pass through the origin because of the -1 term, and it has a positive slope since (2)^x is an increasing function.Evaluate Option (B): Now let's evaluate option (B) g(x) = -((3)/(4))^x + 1. If we take y = -g(x), we get y = -(-((3)/(4))^x + 1) = ((3)/(4))^x - 1. Similar to option (A), this function does not pass through the origin because of the -1 term, and it has a positive slope since ((3)/(4))^x is an increasing function (though less steep than (2)^x).Evaluate Option (C): Next, let's evaluate option (C) g(x) = ((2)/(5))^x - 1. If we take y = -g(x), we get y = -(((2)/(5))^x - 1) = -((2)/(5))^x + 1. This function does not pass through the origin because of the +1 term, and it has a positive slope since -((2)/(5))^x is a decreasing function.Evaluate Option (D): Finally, let's evaluate option (D) g(x) = 3^x. If we take y = -g(x), we get y = -3^x. This function passes through the origin because there is no constant term added or subtracted from 3^x. Also, since 3^x is an increasing function, -3^x will be a decreasing function, which means it will have a negative slope.Final Evaluation: Based on the evaluations, the only function that satisfies both conditions (negative slope and passing through the origin) is option (D) g(x) = 3^x. Therefore, the correct answer is (D).

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