Mister Exam

Graphing y = 3x-1/2x-1

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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             x    
f(x) = 3*x - - - 1
             2    
f(x)=(x2+3x)1f{\left(x \right)} = \left(- \frac{x}{2} + 3 x\right) - 1
f = -x/2 + 3*x - 1
The graph of the function
02468-8-6-4-2-1010-5050
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:
(x2+3x)1=0\left(- \frac{x}{2} + 3 x\right) - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=25x_{1} = \frac{2}{5}
Numerical solution
x1=0.4x_{1} = 0.4
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*x - x/2 - 1.
1+(030)-1 + \left(0 \cdot 3 - 0\right)
The result:
f(0)=1f{\left(0 \right)} = -1
The point:
(0, -1)
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
52=0\frac{5}{2} = 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
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
0=00 = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
limx((x2+3x)1)=\lim_{x \to -\infty}\left(\left(- \frac{x}{2} + 3 x\right) - 1\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
limx((x2+3x)1)=\lim_{x \to \infty}\left(\left(- \frac{x}{2} + 3 x\right) - 1\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 3*x - x/2 - 1, divided by x at x->+oo and x ->-oo
limx((x2+3x)1x)=52\lim_{x \to -\infty}\left(\frac{\left(- \frac{x}{2} + 3 x\right) - 1}{x}\right) = \frac{5}{2}
Let's take the limit
so,
inclined asymptote equation on the left:
y=5x2y = \frac{5 x}{2}
limx((x2+3x)1x)=52\lim_{x \to \infty}\left(\frac{\left(- \frac{x}{2} + 3 x\right) - 1}{x}\right) = \frac{5}{2}
Let's take the limit
so,
inclined asymptote equation on the right:
y=5x2y = \frac{5 x}{2}
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:
(x2+3x)1=5x21\left(- \frac{x}{2} + 3 x\right) - 1 = - \frac{5 x}{2} - 1
- No
(x2+3x)1=5x2+1\left(- \frac{x}{2} + 3 x\right) - 1 = \frac{5 x}{2} + 1
- No
so, the function
not is
neither even, nor odd