Mister Exam

Other calculators

Plotting a function graph/ x-1-ln(x)/x

Graphing y = x-1-ln(x)/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
               log(x)
f(x) = x - 1 - ------
                 x
f(x)=(x−1)−log⁡(x)xf{\left(x \right)} = \left(x - 1\right) - \frac{\log{\left(x \right)}}{x} 
f = x - 1 - log(x)/x
The graph of the function
02468-8-6-4-2-10100100
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
(x−1)−log⁡(x)x=0\left(x - 1\right) - \frac{\log{\left(x \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=1.00000011533972x_{1} = 1.00000011533972
x2=0.99999986420408x_{2} = 0.99999986420408
x3=1.00000008547245x_{3} = 1.00000008547245
x4=1.00000010719113x_{4} = 1.00000010719113
x5=1.00000028592075x_{5} = 1.00000028592075
x6=1.00000012342214x_{6} = 1.00000012342214
x7=0.999999875662866x_{7} = 0.999999875662866
x8=1.00000065036675x_{8} = 1.00000065036675
x9=0.999999874698431x_{9} = 0.999999874698431
x10=0.999999881467175x_{10} = 0.999999881467175
x11=1.00000020393183x_{11} = 1.00000020393183
x12=1.00000015592805x_{12} = 1.00000015592805
x13=1.00000010535343x_{13} = 1.00000010535343
x14=1.00000011317214x_{14} = 1.00000011317214
x15=1.00000009424262x_{15} = 1.00000009424262
x16=1.00000009286636x_{16} = 1.00000009286636
x17=1.00000014025503x_{17} = 1.00000014025503
x18=1.00000019616887x_{18} = 1.00000019616887
x19=0.999999945804162x_{19} = 0.999999945804162
x20=0.99999986323292x_{20} = 0.99999986323292
x21=1.00000014761404x_{21} = 1.00000014761404
x22=1.00000010358587x_{22} = 1.00000010358587
x23=1.0000001654058x_{23} = 1.0000001654058
x24=1.00000016050618x_{24} = 1.00000016050618
x25=1.00000012505702x_{25} = 1.00000012505702
x26=0.9999998940502x_{26} = 0.9999998940502
x27=1.00000012778945x_{27} = 1.00000012778945
x28=0.999999862193662x_{28} = 0.999999862193662
x29=1.0000000136345x_{29} = 1.0000000136345
x30=0.999999865386221x_{30} = 0.999999865386221
x31=0.999999862306678x_{31} = 0.999999862306678
x32=0.999999870471843x_{32} = 0.999999870471843
x33=1.0000001980581x_{33} = 1.0000001980581
x34=1.00000009866416x_{34} = 1.00000009866416
x35=0.999999891076942x_{35} = 0.999999891076942
x36=1.00000048464644x_{36} = 1.00000048464644
x37=0.999999866847997x_{37} = 0.999999866847997
x38=1.00000009713884x_{38} = 1.00000009713884
x39=0.999999892641617x_{39} = 0.999999892641617
x40=0.999999879786295x_{40} = 0.999999879786295
x41=1.00000018901723x_{41} = 1.00000018901723
x42=1.00000009024545x_{42} = 1.00000009024545
x43=0.99999997542134x_{43} = 0.99999997542134
x44=1.00000008436636x_{44} = 1.00000008436636
x45=1.00000013368916x_{45} = 1.00000013368916
x46=1.00000020615424x_{46} = 1.00000020615424
x47=0.999999883130723x_{47} = 0.999999883130723
x48=0.999999923057501x_{48} = 0.999999923057501
x49=1.00000038382559x_{49} = 1.00000038382559
x50=0.999999895489953x_{50} = 0.999999895489953
x51=1.00000023112859x_{51} = 1.00000023112859
x52=1.00000010188429x_{52} = 1.00000010188429
x53=1.000000711697x_{53} = 1.000000711697
x54=1.00000078196751x_{54} = 1.00000078196751
x55=0.999999873015374x_{55} = 0.999999873015374
x56=1.00000017066293x_{56} = 1.00000017066293
x57=1.00000009153471x_{57} = 1.00000009153471
x58=1.00000008899648x_{58} = 1.00000008899648
x59=0.9999998863907x_{59} = 0.9999998863907
x60=0.999999862536128x_{60} = 0.999999862536128
x61=1.00000012245496x_{61} = 1.00000012245496
x62=1.00000024953019x_{62} = 1.00000024953019
x63=0.999999884772927x_{63} = 0.999999884772927
x64=1.00000006235479x_{64} = 1.00000006235479
x65=0.999999889544192x_{65} = 0.999999889544192
x66=1.00000011109535x_{66} = 1.00000011109535
x67=1.00000057750572x_{67} = 1.00000057750572
x68=1.00000010910347x_{68} = 1.00000010910347
x69=1.00000015163989x_{69} = 1.00000015163989
x70=0.999999892579114x_{70} = 0.999999892579114
x71=1.00000011997368x_{71} = 1.00000011997368
x72=0.999999876394517x_{72} = 0.999999876394517
x73=0.999999868192182x_{73} = 0.999999868192182
x74=1.00000008661174x_{74} = 1.00000008661174
x75=0.999999896898342x_{75} = 0.999999896898342
x76=1.00000018241934x_{76} = 1.00000018241934
x77=1.00000022145118x_{77} = 1.00000022145118
x78=1.00000011760454x_{78} = 1.00000011760454
x79=0.999999863003551x_{79} = 0.999999863003551
x80=1.00000008778585x_{80} = 1.00000008778585
x81=0.999999878093228x_{81} = 0.999999878093228
x82=0.999999866728964x_{82} = 0.999999866728964
x83=1.00000009566592x_{83} = 1.00000009566592
x84=1.00000025083693x_{84} = 1.00000025083693
x85=0.999999887981699x_{85} = 0.999999887981699
x86=1.00000021235475x_{86} = 1.00000021235475
x87=1.00000010024489x_{87} = 1.00000010024489
x88=1.00000014382621x_{88} = 1.00000014382621
x89=1.00000013688159x_{89} = 1.00000013688159
x90=0.999999882863149x_{90} = 0.999999882863149
x91=0.999999864447731x_{91} = 0.999999864447731
x92=1.00000069476539x_{92} = 1.00000069476539
x93=1.00000013066287x_{93} = 1.00000013066287
x94=1.0000001763186x_{94} = 1.0000001763186
x95=0.999999871358401x_{95} = 0.999999871358401
x96=1.00000008329199x_{96} = 1.00000008329199
x97=1.00000024097706x_{97} = 1.00000024097706
x98=0.999999905728848x_{98} = 0.999999905728848
x99=0.999999869743854x_{99} = 0.999999869743854
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x - 1 - log(x)/x.
−log⁡(0)0−1- \frac{\log{\left(0 \right)}}{0} - 1
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
1+log⁡(x)x2−1x2=01 + \frac{\log{\left(x \right)}}{x^{2}} - \frac{1}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
The values of the extrema at the points:
(1, 0)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=1x_{1} = 1
The function has no maxima
Decreasing at intervals
[1,∞)\left[1, \infty\right)
Increasing at intervals
(−∞,1]\left(-\infty, 1\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
3−2log⁡(x)x3=0\frac{3 - 2 \log{\left(x \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=e32x_{1} = e^{\frac{3}{2}}
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(3−2log⁡(x)x3)=−∞\lim_{x \to 0^-}\left(\frac{3 - 2 \log{\left(x \right)}}{x^{3}}\right) = -\infty
lim⁡x→0+(3−2log⁡(x)x3)=∞\lim_{x \to 0^+}\left(\frac{3 - 2 \log{\left(x \right)}}{x^{3}}\right) = \infty
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,e32]\left(-\infty, e^{\frac{3}{2}}\right]
Convex at the intervals
[e32,∞)\left[e^{\frac{3}{2}}, \infty\right)
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x−1)−log⁡(x)x)=−∞\lim_{x \to -\infty}\left(\left(x - 1\right) - \frac{\log{\left(x \right)}}{x}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x−1)−log⁡(x)x)=∞\lim_{x \to \infty}\left(\left(x - 1\right) - \frac{\log{\left(x \right)}}{x}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x - 1 - log(x)/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−1)−log⁡(x)xx)=1\lim_{x \to -\infty}\left(\frac{\left(x - 1\right) - \frac{\log{\left(x \right)}}{x}}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞((x−1)−log⁡(x)xx)=1\lim_{x \to \infty}\left(\frac{\left(x - 1\right) - \frac{\log{\left(x \right)}}{x}}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = x
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
(x−1)−log⁡(x)x=−x−1+log⁡(−x)x\left(x - 1\right) - \frac{\log{\left(x \right)}}{x} = - x - 1 + \frac{\log{\left(- x \right)}}{x}
- No
(x−1)−log⁡(x)x=x+1−log⁡(−x)x\left(x - 1\right) - \frac{\log{\left(x \right)}}{x} = x + 1 - \frac{\log{\left(- x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator