Question
Compare the two logarithmic functions f(x)=log3x
𝑓
(
𝑥
)
=
log
3
𝑥
and g(x)=log7x
𝑔
(
𝑥
)
=
log
7
𝑥
. Which statement correctly describes the similarities and differences between these two functions?(1 point)
Responses
Functions f(x)
𝑓
(
𝑥
)
and g(x)
𝑔
(
𝑥
)
will have similar shapes but will have different x-intercepts.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different x -intercepts.
Functions f(x)
𝑓
(
𝑥
)
and g(x)
𝑔
(
𝑥
)
will have similar shapes but will have different vertical asymptotes.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different vertical asymptotes.
Functions f(x)
𝑓
(
𝑥
)
and g(x)
𝑔
(
𝑥
)
will have similar shapes but will have different horizontal asymptotes.
Functions f left parenthesis x right pare
𝑓
(
𝑥
)
=
log
3
𝑥
and g(x)=log7x
𝑔
(
𝑥
)
=
log
7
𝑥
. Which statement correctly describes the similarities and differences between these two functions?(1 point)
Responses
Functions f(x)
𝑓
(
𝑥
)
and g(x)
𝑔
(
𝑥
)
will have similar shapes but will have different x-intercepts.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different x -intercepts.
Functions f(x)
𝑓
(
𝑥
)
and g(x)
𝑔
(
𝑥
)
will have similar shapes but will have different vertical asymptotes.
Functions f left parenthesis x right parenthesis and g left parenthesis x right parenthesis will have similar shapes but will have different vertical asymptotes.
Functions f(x)
𝑓
(
𝑥
)
and g(x)
𝑔
(
𝑥
)
will have similar shapes but will have different horizontal asymptotes.
Functions f left parenthesis x right pare
Answers
GPT-4o mini
To compare the two logarithmic functions \( f(x) = \log_3 x \) and \( g(x) = \log_7 x \), we need to understand their characteristics:
1. **Domain**: Both functions have the same domain, which is \( x > 0 \).
2. **Shape**: Both functions are logarithmic and will have a similar shape because they both increase without bound as \( x \) approaches infinity and decrease toward negative infinity as \( x \) approaches zero.
3. **X-intercepts**: The x-intercepts of both functions occur at \( x = 1 \) because \( \log_b(1) = 0 \) for any positive base \( b \). Therefore, both \( f(x) \) and \( g(x) \) have the same x-intercept at (1, 0).
4. **Y-intercepts**: Both functions do not have y-intercepts as they are not defined for \( x \leq 0 \).
5. **Vertical Asymptotes**: Both functions approach negative infinity as \( x \) approaches zero from the right, which means they both have a vertical asymptote at \( x = 0 \).
6. **Horizontal Asymptotes**: Neither function has horizontal asymptotes as they both increase without bound.
Given these comparisons, the statement that would accurately describe the similarities and differences is:
**Functions \( f(x) \) and \( g(x) \) will have similar shapes but will have different vertical asymptotes.**
This is **not correct**, as there is no difference in vertical asymptotes; both functions share the same behavior.
The accurate statement would be that they have similar shapes and the same x-intercept.
Thus, none of the provided responses are fully correct. A correct modified response would articulate that they have similar shapes and identical x-intercepts.
1. **Domain**: Both functions have the same domain, which is \( x > 0 \).
2. **Shape**: Both functions are logarithmic and will have a similar shape because they both increase without bound as \( x \) approaches infinity and decrease toward negative infinity as \( x \) approaches zero.
3. **X-intercepts**: The x-intercepts of both functions occur at \( x = 1 \) because \( \log_b(1) = 0 \) for any positive base \( b \). Therefore, both \( f(x) \) and \( g(x) \) have the same x-intercept at (1, 0).
4. **Y-intercepts**: Both functions do not have y-intercepts as they are not defined for \( x \leq 0 \).
5. **Vertical Asymptotes**: Both functions approach negative infinity as \( x \) approaches zero from the right, which means they both have a vertical asymptote at \( x = 0 \).
6. **Horizontal Asymptotes**: Neither function has horizontal asymptotes as they both increase without bound.
Given these comparisons, the statement that would accurately describe the similarities and differences is:
**Functions \( f(x) \) and \( g(x) \) will have similar shapes but will have different vertical asymptotes.**
This is **not correct**, as there is no difference in vertical asymptotes; both functions share the same behavior.
The accurate statement would be that they have similar shapes and the same x-intercept.
Thus, none of the provided responses are fully correct. A correct modified response would articulate that they have similar shapes and identical x-intercepts.