Mister Exam
Graphing y = 7-4^(-6*x+2)
The solution
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-6*x + 2 f(x) = 7 - 4
$$f{\left(x \right)} = 7 - 4^{2 - 6 x}$$
f = 7 - 4^(2 - 6*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:
$$7 - 4^{2 - 6 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \log{\left(\left(\frac{16}{7}\right)^{\frac{1}{12 \log{\left(2 \right)}}} \right)}$$
Numerical solution
$$x_{1} = 0.099387089828533$$
so we need to solve the equation:
$$7 - 4^{2 - 6 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \log{\left(\left(\frac{16}{7}\right)^{\frac{1}{12 \log{\left(2 \right)}}} \right)}$$
Numerical solution
$$x_{1} = 0.099387089828533$$
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
$$6 \cdot 4^{2 - 6 x} \log{\left(4 \right)} = 0$$
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
$$6 \cdot 4^{2 - 6 x} \log{\left(4 \right)} = 0$$
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
$$- 576 \cdot 4^{- 6 x} \log{\left(4 \right)}^{2} = 0$$
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
$$- 576 \cdot 4^{- 6 x} \log{\left(4 \right)}^{2} = 0$$
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
$$\lim_{x \to -\infty}\left(7 - 4^{2 - 6 x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(7 - 4^{2 - 6 x}\right) = 7$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 7$$
$$\lim_{x \to -\infty}\left(7 - 4^{2 - 6 x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(7 - 4^{2 - 6 x}\right) = 7$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 7$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 7 - 4^(-6*x + 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{7 - 4^{2 - 6 x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{7 - 4^{2 - 6 x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{7 - 4^{2 - 6 x}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{7 - 4^{2 - 6 x}}{x}\right) = 0$$
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:
$$7 - 4^{2 - 6 x} = 7 - 4^{6 x + 2}$$
- No
$$7 - 4^{2 - 6 x} = 4^{6 x + 2} - 7$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$7 - 4^{2 - 6 x} = 7 - 4^{6 x + 2}$$
- No
$$7 - 4^{2 - 6 x} = 4^{6 x + 2} - 7$$
- No
so, the function
not is
neither even, nor odd