Question
Compare the functions f(x)=4x and g(x)=log4x . The graph of f(x) contains the points (1,4) and (−1, 14) . Select the answer choice that contains two points on the graph of g(x) .(1 point) Responses (1,14) and (−1, 4) open paren 1 comma 1 fourth close paren and (−1, 4) (−1,−4) and (1,−14) open paren negative 1 comma negative 4 close paren and open paren 1 comma negative 1 fourth close paren (1,4) and (−1, 14) open paren 1 comma 4 close paren and (−1, 14) (4,1) and (14, −1)
Answers
GPT-4o mini
To analyze the functions \( f(x) = 4x \) and \( g(x) = \log_4 x \), we first clarify the behavior of \( g(x) \).
The function \( g(x) = \log_4 x \) is defined for \( x > 0 \). For \( g(x) \) to yield real values, \( x \) must be positive. The logarithmic function \( g(x) = \log_4 x \) gives the power to which the base \( 4 \) must be raised to obtain \( x \).
### Evaluating Points on \( g(x) \):
1. **At \( x = 1 \)**:
\[
g(1) = \log_4(1) = 0 \quad (\text{since } 4^0 = 1)
\]
Point: \( (1, 0) \)
2. **At \( x = 4 \)**:
\[
g(4) = \log_4(4) = 1 \quad (\text{since } 4^1 = 4)
\]
Point: \( (4, 1) \)
3. **At \( x = 16 \)**:
\[
g(16) = \log_4(16) = 2 \quad (\text{since } 4^2 = 16)
\]
Point: \( (16, 2) \)
### Compare with Given Response Options:
Given the response options and extracted points, we note:
- **(1, 14)** and **(−1, 4)**
- **(1, \( \frac{1}{4} \))** and **(−1, 4)**
- **(−1, −4)** and **(1, −14)**
- **(−1, −4)** and **(1, \( -\frac{1}{4} \))**
- **(1, 4)** and **(−1, 14)**
- **(4, 1)** and **(14, −1)**
### Conclusion:
The only valid choice with points on \( g(x) \) is:
- **(4, 1)** (since \( g(4) = 1 \))
The equivalent point available in the choices is:
- **(1, \( \frac{1}{4} \))**, which is related to \( g(x) \).
Since none of the given pairs seem to have both correct points from \( g(x) \), but if looked at the context and intention of selecting for \( g(x) \), \( (4, 1) \) is significant.
### Final Answer:
The appropriate response among the given options is **(4, 1) and (14, −1)** as the only points correctly reflecting the \( g(x) \) evaluation deterministically.
The function \( g(x) = \log_4 x \) is defined for \( x > 0 \). For \( g(x) \) to yield real values, \( x \) must be positive. The logarithmic function \( g(x) = \log_4 x \) gives the power to which the base \( 4 \) must be raised to obtain \( x \).
### Evaluating Points on \( g(x) \):
1. **At \( x = 1 \)**:
\[
g(1) = \log_4(1) = 0 \quad (\text{since } 4^0 = 1)
\]
Point: \( (1, 0) \)
2. **At \( x = 4 \)**:
\[
g(4) = \log_4(4) = 1 \quad (\text{since } 4^1 = 4)
\]
Point: \( (4, 1) \)
3. **At \( x = 16 \)**:
\[
g(16) = \log_4(16) = 2 \quad (\text{since } 4^2 = 16)
\]
Point: \( (16, 2) \)
### Compare with Given Response Options:
Given the response options and extracted points, we note:
- **(1, 14)** and **(−1, 4)**
- **(1, \( \frac{1}{4} \))** and **(−1, 4)**
- **(−1, −4)** and **(1, −14)**
- **(−1, −4)** and **(1, \( -\frac{1}{4} \))**
- **(1, 4)** and **(−1, 14)**
- **(4, 1)** and **(14, −1)**
### Conclusion:
The only valid choice with points on \( g(x) \) is:
- **(4, 1)** (since \( g(4) = 1 \))
The equivalent point available in the choices is:
- **(1, \( \frac{1}{4} \))**, which is related to \( g(x) \).
Since none of the given pairs seem to have both correct points from \( g(x) \), but if looked at the context and intention of selecting for \( g(x) \), \( (4, 1) \) is significant.
### Final Answer:
The appropriate response among the given options is **(4, 1) and (14, −1)** as the only points correctly reflecting the \( g(x) \) evaluation deterministically.