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

(1,8)
and (−1,0.125)
left parenthesis 1 comma 8 right parenthesis and left parenthesis negative 1 comma 0.125 right parenthesis

(8,1)
and (0.125,−1)

Answer 1

To compare the functions \( f(x) = 8x \) and \( g(x) = \log_8(x) \), we need to evaluate the points from the function \( f(x) \) and check their correlating points on \( g(x) \).

1. **Evaluate \( f(x) \)**:
- When \( x = 1 \):
\[ f(1) = 8 \cdot 1 = 8 \]
- When \( x = -1 \):
\[ f(-1) = 8 \cdot (-1) = -8 \]

The points given for \( f(x) \) are \( (1, 8) \) and \( (-1, 0.125) \).

2. **Evaluate \( g(x) \)**:
\( g(x) = \log_8(x) \), which is the inverse of \( f(x) \).
- To find the points on \( g(x) \), we can use the points from \( f(x) \) since they are inverses:
- For the point \( (1, 8) \) from \( f(x) \):
\[ g(8) = \log_8(8) = 1 \]
So, the point is \( (8, 1) \).
- For the point \( (-1, 0.125) \) from \( f(x) \):
\[ g(0.125) = \log_8(0.125) \]
Since \( 0.125 = 8^{-1} \), we have:
\[ g(0.125) = -1 \]
Therefore, the point is \( (0.125, -1) \).

Now we summarize the correlating points we found:
- From \( (1, 8) \), we get \( (8, 1) \).
- From \( (-1, 0.125) \), we get \( (0.125, -1) \).

3. **Select the correct answer**:
By analyzing the choices provided:

- **(−1,−8) and (1,−0.125)** - Incorrect.
- **(1,0.125) and (−1,8)** - Incorrect.
- **(1,8) and (−1,0.125)** - Incorrect.
- **(8,1) and (0.125,−1)** - Correct!

Thus, the correct choice is **(8, 1) and (0.125, -1)**.