f(x)=logx & g(x)=x,so i need graph of the function f(x)/g(x)Please describe these key features: intercepts, asymptotes, domain, range, symmetry, maxima, minima, slope.
You need to know the parts of the graph y = (logx)/x.
If you are allowed to use your calculator, you can find intercepts, maxima, minima, all on there. If not:
- y intercept is when x = 0, but since this is log, there is an asymptote at x = 0, so no y-intercept.
- x-intercept is when y = 0, so if 0=logx/x, 0=logx, and x = 1 (x-intercept is 1)
Asymptotes: vertical asymptote at x = 0 (y-axis)
Domain: x>0 (domain of logx)
Range: all real numbers (range of logx)
Symmetry: none, the function does not cross the y-axis, so it can't have y-axis or origin symmetry
Maxima and Minima: the only way I know to do this without the calculator is by calculus. Take the derivative and you get (1-logx)/x^2, and relative extrema are where 1-logx = 0, so logx = 1, so x = 10. This is a maximum because the function is increasing until there, decreasing after that point.
Slope: increasing until x = 10, then decreasing.
The equation of the graph is y= logx/x
To find the intercepts:
To find y-intercept, sub. x=0
y= log (0) /0.
Since there is an asymptote when x=0, there is no y-intersect as curve didn't touch y-axis
To find x-intercept, sub.y=0
0= logx/ x
Thus, x-intercept is one.
2) There is a vertical asymptote when x equals zero
3) The domain is x is more than zero
4) The range of this function is more than zero, and must be real numbers too, not negative or complex numerals
5) sadly, no symmetry involved here. The curve didn't cross y-axis at all
6) The maximum value of this curve is x=10, finding it using the scientific calculator
7) Slope increases until it reaches x=10 and then it falls after passing x=10