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  • Graphing y =:
  • -x^4+4x
  • x^3-2*x^2+x+3 x^3-2*x^2+x+3
  • x^3+3x^2-9x+4
  • x^3+2x
  • Identical expressions

  • tan(three *x-pi/ four)
  • tangent of (3 multiply by x minus Pi divide by 4)
  • tangent of (three multiply by x minus Pi divide by four)
  • tan(3x-pi/4)
  • tan3x-pi/4
  • tan(3*x-pi divide by 4)
  • Similar expressions

  • tan(3*x+pi/4)

Graphing y = tan(3*x-pi/4)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /      pi\
f(x) = tan|3*x - --|
          \      4 /
$$f{\left(x \right)} = \tan{\left(3 x - \frac{\pi}{4} \right)}$$
f = tan(3*x - pi/4)
The graph of the function
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:
$$\tan{\left(3 x - \frac{\pi}{4} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = \frac{\pi}{12}$$
Numerical solution
$$x_{1} = 97.6511716490827$$
$$x_{2} = -79.3252145031423$$
$$x_{3} = -85.6083998103219$$
$$x_{4} = -1.83259571459405$$
$$x_{5} = 0.261799387799149$$
$$x_{6} = 55.7632696012188$$
$$x_{7} = 84.037603483527$$
$$x_{8} = -32.2013246992954$$
$$x_{9} = 24.3473430653209$$
$$x_{10} = 52.621676947629$$
$$x_{11} = 15.9697626557481$$
$$x_{12} = 46.3384916404494$$
$$x_{13} = -74.0892267471593$$
$$x_{14} = 13.8753675533549$$
$$x_{15} = -25.9181393921158$$
$$x_{16} = -52.0980781720307$$
$$x_{17} = 72.5184304203644$$
$$x_{18} = -100.269165527074$$
$$x_{19} = -19.6349540849362$$
$$x_{20} = -76.1836218495525$$
$$x_{21} = 68.329640215578$$
$$x_{22} = -67.8060414399797$$
$$x_{23} = 81.9432083811338$$
$$x_{24} = 96.6039740978861$$
$$x_{25} = 6.54498469497874$$
$$x_{26} = -91.8915851175014$$
$$x_{27} = 44.2440965380563$$
$$x_{28} = 4.45058959258554$$
$$x_{29} = -13.3517687777566$$
$$x_{30} = -83.5140047079287$$
$$x_{31} = 50.5272818452358$$
$$x_{32} = -81.4196096055355$$
$$x_{33} = 9.68657734856853$$
$$x_{34} = 88.2263936883134$$
$$x_{35} = -17.540558982543$$
$$x_{36} = 26.4417381677141$$
$$x_{37} = -59.4284610304069$$
$$x_{38} = 42.1497014356631$$
$$x_{39} = 28.5361332701073$$
$$x_{40} = 94.5095789954929$$
$$x_{41} = 77.7544181763474$$
$$x_{42} = -47.9092879672443$$
$$x_{43} = 79.8488132787406$$
$$x_{44} = -65.7116463375865$$
$$x_{45} = 33.7721210260903$$
$$x_{46} = -98.174770424681$$
$$x_{47} = 59.9520598060052$$
$$x_{48} = 20.1585528605345$$
$$x_{49} = -28.012534494509$$
$$x_{50} = -23.8237442897226$$
$$x_{51} = -57.3340659280137$$
$$x_{52} = 8.63937979737193$$
$$x_{53} = -8.11578102177363$$
$$x_{54} = 40.0553063332699$$
$$x_{55} = -69.9004365423729$$
$$x_{56} = 31.6777259236971$$
$$x_{57} = 53.6688744988256$$
$$x_{58} = -54.1924732744239$$
$$x_{59} = -56.2868683768171$$
$$x_{60} = 90.3207887907066$$
$$x_{61} = 74.6128255227576$$
$$x_{62} = 35.8665161284835$$
$$x_{63} = -34.2957198016886$$
$$x_{64} = 62.0464549083984$$
$$x_{65} = -30.1069295969022$$
$$x_{66} = -21.7293491873294$$
$$x_{67} = -12.30457122656$$
$$x_{68} = -89.7971900151083$$
$$x_{69} = -87.7027949127151$$
$$x_{70} = -3.92699081698724$$
$$x_{71} = -71.9948316447661$$
$$x_{72} = 57.857664703612$$
$$x_{73} = 99.7455667514759$$
$$x_{74} = 75.6600230739542$$
$$x_{75} = -15.4461638801498$$
$$x_{76} = -63.6172512351933$$
$$x_{77} = -61.5228561328001$$
$$x_{78} = -39.5317075576716$$
$$x_{79} = 37.9609112308767$$
$$x_{80} = -43.720497762458$$
$$x_{81} = -45.8148928648512$$
$$x_{82} = 48.4328867428426$$
$$x_{83} = -96.0803753222878$$
$$x_{84} = 2.35619449019234$$
$$x_{85} = -6.02138591938044$$
$$x_{86} = 11.7809724509617$$
$$x_{87} = 66.2352451131848$$
$$x_{88} = -10.2101761241668$$
$$x_{89} = 18.0641577581413$$
$$x_{90} = -37.4373124552784$$
$$x_{91} = 86.1319985859202$$
$$x_{92} = -50.0036830696375$$
$$x_{93} = -41.6261026600648$$
$$x_{94} = 30.6305283725005$$
$$x_{95} = 64.1408500107916$$
$$x_{96} = -78.2780169519457$$
$$x_{97} = -35.3429173528852$$
$$x_{98} = 22.2529479629277$$
$$x_{99} = 70.4240353179712$$
$$x_{100} = 92.4151838930998$$
$$x_{101} = -93.9859802198946$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(3*x - pi/4).
$$\tan{\left(- \frac{\pi}{4} + 0 \cdot 3 \right)}$$
The result:
$$f{\left(0 \right)} = -1$$
The point:
(0, -1)
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$3 \tan^{2}{\left(3 x - \frac{\pi}{4} \right)} + 3 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 18 \left(\cot^{2}{\left(3 x + \frac{\pi}{4} \right)} + 1\right) \cot{\left(3 x + \frac{\pi}{4} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\pi}{12}$$

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
$$\left[\frac{\pi}{12}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{\pi}{12}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \tan{\left(3 x - \frac{\pi}{4} \right)}$$
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \tan{\left(3 x - \frac{\pi}{4} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(3*x - pi/4), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x - \frac{\pi}{4} \right)}}{x}\right)$$
True

Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x - \frac{\pi}{4} \right)}}{x}\right)$$
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:
$$\tan{\left(3 x - \frac{\pi}{4} \right)} = - \tan{\left(3 x + \frac{\pi}{4} \right)}$$
- No
$$\tan{\left(3 x - \frac{\pi}{4} \right)} = \tan{\left(3 x + \frac{\pi}{4} \right)}$$
- No
so, the function
not is
neither even, nor odd