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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 1 comma 8 right parenthesis and left parenthesis negative 1 comma 0.125 right parenthesis

(−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)

1 answer

To compare the functions \( f(x) = 8x \) and \( g(x) = \log_8(x) \), we’ll first evaluate \( g(x) \) at the points provided in the answer choices.

  1. For \( x = 1 \):

    • \( g(1) = \log_8(1) = 0 \) (because \( 8^0 = 1 \))

  2. For \( x = -1 \):

    • The logarithm \( \log_8(-1) \) is not defined in the real number system, so this point cannot be evaluated.

  3. For \( x = 8 \):

    • \( g(8) = \log_8(8) = 1 \) (because \( 8^1 = 8 \))

  4. For \( x = 0.125 \):

    • \( g(0.125) = \log_8(0.125) = \log_8\left(\frac{1}{8}\right) = -1 \) (because \( 8^{-1} = 0.125 \))

Now let's review the answer choices based on these evaluations:

  • (1, 8) and (−1, 0.125): These points correspond to \( f(x) \), not \( g(x) \) (and \(-1\) is not valid).

  • (−1,−8) and (1,−0.125): Both points are invalid for \( g(x) \); \(-1\) is not a valid input for log function, and neither of them relates to what \( g(x) \) provides.

  • (1, 0.125) and (−1, 8): \( g(1) = 0 \), so this pair includes an invalid point from the logarithm perspective.

  • (8, 1) and (0.125, −1): Here we get valid points: \( g(8) = 1 \) and \( g(0.125) = -1 \).

Hence, the correct choice that contains two points on the graph of \( g(x) = \log_8(x) \) is: (8, 1) and (0.125, −1).