Question

the graph of $f(x) = 6(0.25)^x$ and its reflection across the $y$-axis, $g(x)$, are shown. what is the domain of $g(x)$? all real numbers greater than or equal to 0 all real numbers less than 0 all real numbers all real numbers greater than 0

Explanation:

Step1: Recall reflection over y - axis

For a function \(y = f(x)\), its reflection over the \(y\) - axis is given by \(g(x)=f(-x)\). Here, \(f(x)=6(0.25)^{x}\), so \(g(x)=f(-x)=6(0.25)^{-x}\). We can rewrite \((0.25)^{-x}\) as \((\frac{1}{0.25})^{x}=4^{x}\), so \(g(x) = 6(4)^{x}\), which is an exponential function.

Step2: Determine the domain of exponential functions

The domain of an exponential function of the form \(y = a\cdot b^{x}\) (where \(a
eq0\) and \(b>0,b
eq1\)) is all real numbers. This is because we can substitute any real number for \(x\) and the function will be defined. Looking at the graph of \(g(x)\), we can see that the graph extends infinitely to the left and right along the \(x\) - axis, which means there are no restrictions on the values of \(x\) that we can plug into \(g(x)\).

Answer:

all real numbers (the option "all real numbers")