Mister Exam
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-
How to use it?
- Graphing y =:
- x^4-3x^2
- x^3-3x^2-9x+10
- x^3-27x
- -x^2+x-6
- Derivative of:
-
(x-1)/(2*x+3)
- How do you in partial fractions?:
- (x-1)/(2*x+3)
-
Identical expressions
- (x- one)/(two *x+ three)
- (x minus 1) divide by (2 multiply by x plus 3)
- (x minus one) divide by (two multiply by x plus three)
- (x-1)/(2x+3)
- x-1/2x+3
- (x-1) divide by (2*x+3)
-
Similar expressions
- (x-1)/(2*x-3)
- (x+1)/(2*x+3)
Graphing y = (x-1)/(2*x+3)
The solution
You have entered
[src]
x - 1
f(x) = -------
2*x + 3
$$f{\left(x \right)} = \frac{x - 1}{2 x + 3}$$
f = (x - 1)/(2*x + 3)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1.5$$
$$x_{1} = -1.5$$
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:
$$\frac{x - 1}{2 x + 3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\frac{x - 1}{2 x + 3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$- \frac{2 \left(x - 1\right)}{\left(2 x + 3\right)^{2}} + \frac{1}{2 x + 3} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \left(x - 1\right)}{\left(2 x + 3\right)^{2}} + \frac{1}{2 x + 3} = 0$$
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
$$\frac{4 \left(\frac{2 \left(x - 1\right)}{2 x + 3} - 1\right)}{\left(2 x + 3\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{4 \left(\frac{2 \left(x - 1\right)}{2 x + 3} - 1\right)}{\left(2 x + 3\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -1.5$$
$$x_{1} = -1.5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 1}{2 x + 3}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{2 x + 3}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{2}$$
$$\lim_{x \to -\infty}\left(\frac{x - 1}{2 x + 3}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{2 x + 3}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x - 1)/(2*x + 3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 1}{x \left(2 x + 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - 1}{x \left(2 x + 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x - 1}{x \left(2 x + 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - 1}{x \left(2 x + 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\frac{x - 1}{2 x + 3} = \frac{- x - 1}{3 - 2 x}$$
- No
$$\frac{x - 1}{2 x + 3} = - \frac{- x - 1}{3 - 2 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x - 1}{2 x + 3} = \frac{- x - 1}{3 - 2 x}$$
- No
$$\frac{x - 1}{2 x + 3} = - \frac{- x - 1}{3 - 2 x}$$
- No
so, the function
not is
neither even, nor odd