Mister Exam
Graphing y = (1/3)^x-2
The solution
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-x f(x) = 3 - 2
$$f{\left(x \right)} = -2 + \left(\frac{1}{3}\right)^{x}$$
f = -2 + (1/3)^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:
$$-2 + \left(\frac{1}{3}\right)^{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\log{\left(2 \right)}}{\log{\left(3 \right)}}$$
Numerical solution
$$x_{1} = -0.630929753571457$$
$$x_{2} = -0.630929753571777$$
so we need to solve the equation:
$$-2 + \left(\frac{1}{3}\right)^{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\log{\left(2 \right)}}{\log{\left(3 \right)}}$$
Numerical solution
$$x_{1} = -0.630929753571457$$
$$x_{2} = -0.630929753571777$$
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
$$\lim_{x \to -\infty}\left(-2 + \left(\frac{1}{3}\right)^{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-2 + \left(\frac{1}{3}\right)^{x}\right) = -2$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -2$$
$$\lim_{x \to -\infty}\left(-2 + \left(\frac{1}{3}\right)^{x}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-2 + \left(\frac{1}{3}\right)^{x}\right) = -2$$
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 (1/3)^x - 2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-2 + \left(\frac{1}{3}\right)^{x}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{-2 + \left(\frac{1}{3}\right)^{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-2 + \left(\frac{1}{3}\right)^{x}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{-2 + \left(\frac{1}{3}\right)^{x}}{x}\right) = 0$$
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:
$$-2 + \left(\frac{1}{3}\right)^{x} = \left(\frac{1}{3}\right)^{- x} - 2$$
- No
$$-2 + \left(\frac{1}{3}\right)^{x} = 2 - \left(\frac{1}{3}\right)^{- x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-2 + \left(\frac{1}{3}\right)^{x} = \left(\frac{1}{3}\right)^{- x} - 2$$
- No
$$-2 + \left(\frac{1}{3}\right)^{x} = 2 - \left(\frac{1}{3}\right)^{- x}$$
- No
so, the function
not is
neither even, nor odd