graph the function ( g(x)=left(\frac{1}{2}\right)^{x} ). which features are correctly stated?\na x…

graph the function ( g(x)=left(\frac{1}{2}\right)^{x} ). which features are correctly stated?\na x - intercept: none\nb y - intercept: ( (0,1) )\nc asymptote: ( y = 0 )\nd as ( x \to infty, f(x) \to infty )\ne as ( x \to -infty, f(x) \to 0 )

graph the function ( g(x)=left(\frac{1}{2}\right)^{x} ). which features are correctly stated?\na x - intercept: none\nb y - intercept: ( (0,1) )\nc asymptote: ( y = 0 )\nd as ( x \to infty, f(x) \to infty )\ne as ( x \to -infty, f(x) \to 0 )

Answer

Answer:

A. x - intercept: none, B. y - intercept: (0,1), C. asymptote: y = 0

Explanation:

Step1: Find the x - intercept

Set (y = g(x)=0), so (\left(\frac{1}{2}\right)^{x}=0). Since for any real number (x), (a^{x}>0) ((a>0,a\neq1)), here (a = \frac{1}{2}), there is no solution for (\left(\frac{1}{2}\right)^{x}=0). So, the x - intercept is none.

Step2: Find the y - intercept

Set (x = 0), then (g(0)=\left(\frac{1}{2}\right)^{0}). By the rule (a^{0}=1) ((a\neq0)), we have (g(0)=1). So, the y - intercept is ((0,1)).

Step3: Find the horizontal asymptote

For the exponential function (y = a^{x}) ((a>0,a\neq1)), when (|x|\to+\infty): As (x\to+\infty), (y=\left(\frac{1}{2}\right)^{x}=\frac{1}{2^{x}}\to0). The horizontal asymptote of (y = a^{x}) ((0 < a<1)) is (y = 0).

Step4: Analyze the end - behavior

As (x\to+\infty), (g(x)=\left(\frac{1}{2}\right)^{x}=\frac{1}{2^{x}}\to0) (not (\infty)). As (x\to-\infty), (g(x)=\left(\frac{1}{2}\right)^{x}=2^{-x}=2^{|x|}\to+\infty) (not (0)).