To compare the graphs of the functions [tex]\(f(x) = \ln(x)\)[/tex] and [tex]\(g(x) = -5 \ln(x)\)[/tex], we'll analyze each characteristic point by point:
1. Vertical Asymptote:
- The natural logarithm function, [tex]\(\ln(x)\)[/tex], has a vertical asymptote at [tex]\(x = 0\)[/tex]. This means that as [tex]\(x\)[/tex] approaches 0 from the right-hand side, the function values tend to negative infinity.
- Since [tex]\(g(x) = -5 \ln(x)\)[/tex], it inherits the same vertical asymptote as [tex]\(\ln(x)\)[/tex], which is at [tex]\(x = 0\)[/tex].
2. Y-Intercept:
- For [tex]\(f(x) = \ln(x)\)[/tex], there is no y-intercept because [tex]\(\ln(x)\)[/tex] is undefined at [tex]\(x = 0\)[/tex].
- For [tex]\(g(x) = -5 \ln(x)\)[/tex], we can check if there's a y-intercept by setting [tex]\(x = 1\)[/tex]:
[tex]\[
g(1) = -5 \ln(1) = -5 \cdot 0 = 0
\][/tex]
So, [tex]\(g(x)\)[/tex] crosses the y-axis at [tex]\((1, 0)\)[/tex], which means it has a y-intercept.
3. Domain:
- The domain of [tex]\(f(x) = \ln(x)\)[/tex] is [tex]\(x > 0\)[/tex], meaning [tex]\(x\)[/tex] can be any positive real number.
- The statement that the domain of [tex]\(g(x) = -5 \ln(x)\)[/tex] is [tex]\(\{x \mid -5 < x < \infty\}\)[/tex] is incorrect. The correct domain of [tex]\(g(x)\)[/tex] is also [tex]\(x > 0\)[/tex] because it is a logarithmic function, and logarithms are only defined for positive [tex]\(x\)[/tex] values.
4. Behavior as [tex]\(x\)[/tex] Increases:
- The function [tex]\(f(x) = \ln(x)\)[/tex] increases as [tex]\(x\)[/tex] increases.
- The function [tex]\(g(x) = -5 \ln(x)\)[/tex] decreases as [tex]\(x\)[/tex] increases because the negative sign and the multiplication by 5 invert and stretch the original function's growth, making it decrease.
5. Transformation:
- The function [tex]\(g(x) = -5 \ln(x)\)[/tex] can be seen as a transformed version of [tex]\(f(x) = \ln(x)\)[/tex].
- When you reflect [tex]\(\ln(x)\)[/tex] over the x-axis, you get [tex]\(-\ln(x)\)[/tex].
- Multiplying by 5 further stretches it vertically by a factor of 5.
- Hence, [tex]\(g(x) = -5 \ln(x)\)[/tex] is the graph of [tex]\(f(x) = \ln(x)\)[/tex] reflected over the x-axis and then stretched vertically by a factor of 5.
Based on the detailed analysis:
- Both functions have a vertical asymptote at [tex]\(x = 0\)[/tex].
- The graph of function [tex]\(g\)[/tex] has a y-intercept at [tex]\((1,0)\)[/tex].
- The statement about the domain [tex]\(\{x \mid -5 < x < \infty\}\)[/tex] is incorrect; the actual domain of [tex]\(g(x)\)[/tex] is [tex]\(x > 0\)[/tex].
- The graph of function [tex]\(g\)[/tex] decreases as [tex]\(x\)[/tex] increases.
- The graph of function [tex]\(g\)[/tex] is the graph of function [tex]\(f\)[/tex] reflected over the x-axis and stretched vertically by a factor of 5.
