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