Question
Compare the two logarithmic functions f(x)=log4x
and g(x)=log9x
. Which statement correctly describes a feature shared by these functions?(1 point)
Responses
The functions f(x)
and g(x)
both decrease as x→∞
.
The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis both decrease as x right arrow infinity .
The functions f(x)
and g(x)
both have a horizontal asymptote at y=0
.
The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis both have a horizontal asymptote at y equals 0 .
The functions f(x)
and g(x)
both have a y-intercept at (0,1)
.
The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis both have a y -intercept at left parenthesis 0 comma 1 right parenthesis .
The functions f(x)
and g(x)
both have a vertical asymptote at x=0
.
and g(x)=log9x
. Which statement correctly describes a feature shared by these functions?(1 point)
Responses
The functions f(x)
and g(x)
both decrease as x→∞
.
The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis both decrease as x right arrow infinity .
The functions f(x)
and g(x)
both have a horizontal asymptote at y=0
.
The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis both have a horizontal asymptote at y equals 0 .
The functions f(x)
and g(x)
both have a y-intercept at (0,1)
.
The functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis both have a y -intercept at left parenthesis 0 comma 1 right parenthesis .
The functions f(x)
and g(x)
both have a vertical asymptote at x=0
.
Answers
GPT-4o mini
To analyze the functions \( f(x) = \log_4 x \) and \( g(x) = \log_9 x \), let's examine the choices given:
1. **The functions \( f(x) \) and \( g(x) \) both decrease as \( x \to \infty \).**
- This statement is false. Both logarithmic functions increase without bound as \( x \) approaches infinity.
2. **The functions \( f(x) \) and \( g(x) \) both have a horizontal asymptote at \( y = 0 \).**
- This statement is true. As \( x \) approaches 0 from the right, both \( \log_4 x \) and \( \log_9 x \) approach negative infinity, which implies that they have horizontal asymptotes at \( y = 0 \) as \( x \) approaches infinity.
3. **The functions \( f(x) \) and \( g(x) \) both have a y-intercept at \( (0,1) \).**
- This statement is false. Both functions do not have a y-intercept, as logarithmic functions are not defined at \( x = 0 \).
4. **The functions \( f(x) \) and \( g(x) \) both have a vertical asymptote at \( x = 0 \).**
- This statement is true. Both logarithmic functions have a vertical asymptote at \( x = 0 \), as they approach negative infinity when \( x \) approaches 0 from the right.
The correct answer based on the shared features of the functions is:
**The functions \( f(x) \) and \( g(x) \) both have a vertical asymptote at \( x = 0 \).**
1. **The functions \( f(x) \) and \( g(x) \) both decrease as \( x \to \infty \).**
- This statement is false. Both logarithmic functions increase without bound as \( x \) approaches infinity.
2. **The functions \( f(x) \) and \( g(x) \) both have a horizontal asymptote at \( y = 0 \).**
- This statement is true. As \( x \) approaches 0 from the right, both \( \log_4 x \) and \( \log_9 x \) approach negative infinity, which implies that they have horizontal asymptotes at \( y = 0 \) as \( x \) approaches infinity.
3. **The functions \( f(x) \) and \( g(x) \) both have a y-intercept at \( (0,1) \).**
- This statement is false. Both functions do not have a y-intercept, as logarithmic functions are not defined at \( x = 0 \).
4. **The functions \( f(x) \) and \( g(x) \) both have a vertical asymptote at \( x = 0 \).**
- This statement is true. Both logarithmic functions have a vertical asymptote at \( x = 0 \), as they approach negative infinity when \( x \) approaches 0 from the right.
The correct answer based on the shared features of the functions is:
**The functions \( f(x) \) and \( g(x) \) both have a vertical asymptote at \( x = 0 \).**