Question

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)

Answer 1

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)**.