Compare the functions f(x)=8x and g(x)=log8x . The graph of f(x) contains the points (1,8) and (−1,0.125) . Select the answer choice that contains two points on the graph of g(x) .(1 point) Responses (−1,−8) and (1,−0.125) left parenthesis negative 1 comma negative 8 right parenthesis and left parenthesis 1 comma negative 0.125 right parenthesis (1,0.125) and (−1,8) left parenthesis 1 comma 0.125 right parenthesis and left parenthesis negative 1 comma 8 right parenthesis (8,1) and (0.125,−1) left parenthesis 8 comma 1 right parenthesis and left parenthesis 0.125 comma negative 1 right parenthesis (1,8) and (−1,0.125)

Question

GPT-4o mini · Answer

To compare the functions $ f(x) = 8x $ and $ g(x) = \log_8 x $, we first evaluate the second function $ g(x) $ at the points provided for $ f(x) $. The function $ g(x) = \log_8 x $ means we will find the logarithm of $ x $ with base 8. 1. For $ f(1) = 8 $: $$ g(8) = \log_8 8 = 1 $$ So, the point $ (8, 1) $ is on the graph of $ g(x) $. 2. For $ f(-1) $: The value $ x = -1 $ is not in the domain of $ g(x) $ because the logarithm function is undefined for negative numbers. Therefore, we cannot find $ g(-1) $. Now let’s look through the provided options to find valid points for $ g(x) $: - **(−1,−8) and (1,−0.125)**: Invalid as $ g(-1) $ is undefined, but $ g(1) = 0 $. - **(1,0.125) and (−1,8)**: $ g(1) = 0 $ which is valid, but $ g(-1) $ is still invalid. - **(8,1) and (0.125,−1)**: $ g(8) = 1 $ is valid, and $ g(0.125) = \log_8(0.125) = \log_8(8^{-1}) = -1$, which is also valid. - **(1,8) and (−1,0.125)**: Invalid, $ g(1) = 0 $ and $ g(-1) $ is undefined. Therefore, the two valid points for the graph of $ g(x) $ are: **(8, 1) and (0.125, -1)**. So the correct answer is: **(8,1) and (0.125,−1)**.