Mister Exam
Graphing y = 3e^(-2x)-4
The solution
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-2*x f(x) = 3*E - 4
$$f{\left(x \right)} = -4 + 3 e^{- 2 x}$$
f = -4 + 3*E^(-2*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:
$$-4 + 3 e^{- 2 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \log{\left(2 \right)} + \frac{\log{\left(3 \right)}}{2}$$
Numerical solution
$$x_{1} = -0.14384103622589$$
so we need to solve the equation:
$$-4 + 3 e^{- 2 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \log{\left(2 \right)} + \frac{\log{\left(3 \right)}}{2}$$
Numerical solution
$$x_{1} = -0.14384103622589$$
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
$$- 6 e^{- 2 x} = 0$$
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
$$- 6 e^{- 2 x} = 0$$
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
$$12 e^{- 2 x} = 0$$
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
$$12 e^{- 2 x} = 0$$
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
$$\lim_{x \to -\infty}\left(-4 + 3 e^{- 2 x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-4 + 3 e^{- 2 x}\right) = -4$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -4$$
$$\lim_{x \to -\infty}\left(-4 + 3 e^{- 2 x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-4 + 3 e^{- 2 x}\right) = -4$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -4$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*E^(-2*x) - 4, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-4 + 3 e^{- 2 x}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{-4 + 3 e^{- 2 x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-4 + 3 e^{- 2 x}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{-4 + 3 e^{- 2 x}}{x}\right) = 0$$
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:
$$-4 + 3 e^{- 2 x} = 3 e^{2 x} - 4$$
- No
$$-4 + 3 e^{- 2 x} = 4 - 3 e^{2 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-4 + 3 e^{- 2 x} = 3 e^{2 x} - 4$$
- No
$$-4 + 3 e^{- 2 x} = 4 - 3 e^{2 x}$$
- No
so, the function
not is
neither even, nor odd