The function y=g(x) where g(x)=3((1)/(2))^(-x)+2 is graphed in the xy-plane. Which of the following is a true statement? Choose 1 answer: (A) The graph of function g is always increasing. (B) The y-intercept of the graph of function g is (0,2). (c) The x-intercept of the graph of function g is (0,3). (D) Function g is symmetric with respect to the y-axis.

Question

The function  y=g(x) where  g(x)=3((1)/(2))^(-x)+2 is graphed in the  xy-plane. Which of the following is a true statement? Choose 1 answer: (A) The graph of function  g is always increasing. (B) The  y-intercept of the graph of function  g is  (0,2). (c) The  x-intercept of the graph of function  g is  (0,3). (D) Function  g is symmetric with respect to the  y-axis.

Bytelearn · Accepted Answer

Analyze function behavior: Analyze the function g(x) to determine its behavior. The function g(x) = 3((1)/(2))^(-x) + 2 is an exponential function with a base of (1/2) raised to the power of -x, and then multiplied by 3 and added to 2. The negative exponent indicates that as x increases, the value of ((1)/(2))^(-x) increases because (1/2)^(-x) is the same as 2^x. This means the function is increasing as x increases.Find y-intercept: Determine the y-intercept of the function g(x). To find the y-intercept, we set x to 0 in the function g(x). g(0) = 3((1)/(2))^(-0) + 2 g(0) = 3(1) + 2 g(0) = 3 + 2 g(0) = 5 The y-intercept of the graph of function g is (0,5), not (0,2).Find x-intercept: Determine the x-intercept of the function g(x). To find the x-intercept, we set g(x) to 0 and solve for x. 0 = 3((1)/(2))^(-x) + 2 -2 = 3((1)/(2))^(-x) -2/3 = ((1)/(2))^(-x) This equation does not have a real solution because ((1)/(2))^(-x) is always positive, and therefore cannot equal -2/3. Hence, there is no x-intercept.Check for y-axis symmetry: Check for symmetry with respect to the y-axis. A function is symmetric with respect to the y-axis if g(-x) = g(x) for all x. Let's check if this is true for g(x). g(-x) = 3((1)/(2))^(-(-x)) + 2 g(-x) = 3((1)/(2))^(x) + 2 Since 3((1)/(2))^(-x) + 2 is not equal to 3((1)/(2))^(x) + 2 for all x, the function g is not symmetric with respect to the y-axis.

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