Mister Exam
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- The intersection points of functions
-
How to use it?
- Graphing y =:
-
|x^2+x-2|
- x(2-x)^2
- x^2-6x+9
- (x+1)*(x-2)^2
-
Identical expressions
- two *x- two -e^(-x)
- 2 multiply by x minus 2 minus e to the power of ( minus x)
- two multiply by x minus two minus e to the power of ( minus x)
- 2*x-2-e(-x)
- 2*x-2-e-x
- 2x-2-e^(-x)
- 2x-2-e(-x)
- 2x-2-e-x
- 2x-2-e^-x
-
Similar expressions
- 2*x+2-e^(-x)
- 2*x-2+e^(-x)
- 2*x-2-e^(x)
Graphing y = 2*x-2-e^(-x)
The solution
You have entered
[src]
-x f(x) = 2*x - 2 - e
$$f{\left(x \right)} = 2 x - 2 - e^{- x}$$
f = 2*x - 1*2 - 1/E^x
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:
$$2 x - 2 - e^{- x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = W\left(\frac{1}{2 e}\right) + 1$$
Numerical solution
$$x_{1} = 1.15718495148381$$
so we need to solve the equation:
$$2 x - 2 - e^{- x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = W\left(\frac{1}{2 e}\right) + 1$$
Numerical solution
$$x_{1} = 1.15718495148381$$
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
$$2 + e^{- x} = 0$$
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
$$2 + e^{- x} = 0$$
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
$$- e^{- x} = 0$$
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
$$- e^{- x} = 0$$
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
$$\lim_{x \to -\infty}\left(2 x - 2 - e^{- x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(2 x - 2 - e^{- x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(2 x - 2 - e^{- x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(2 x - 2 - e^{- x}\right) = \infty$$
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 2*x - 1*2 - 1/E^x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 x - 2 - e^{- x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 x - 2 - e^{- x}}{x}\right) = 2$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 2 x$$
$$\lim_{x \to -\infty}\left(\frac{2 x - 2 - e^{- x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{2 x - 2 - e^{- x}}{x}\right) = 2$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 2 x$$
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:
$$2 x - 2 - e^{- x} = - 2 x - e^{x} - 2$$
- No
$$2 x - 2 - e^{- x} = 2 x + e^{x} + 2$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 x - 2 - e^{- x} = - 2 x - e^{x} - 2$$
- No
$$2 x - 2 - e^{- x} = 2 x + e^{x} + 2$$
- No
so, the function
not is
neither even, nor odd
The graph