Mister Exam

Lang:

Graphing y = (-1)^x

You have entered [src]

           x
f(x) = (-1)

f{\left(x \right)} = \left(-1\right)^{x}

f = (-1)^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:

\left(-1\right)^{x} = 0

Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to (-1)^x.

\left(-1\right)^{0}

The result:

f{\left(0 \right)} = 1

The point:

(0, 1)

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)} =

\left(-1\right)^{x} i \pi = 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)} =

- \left(-1\right)^{x} \pi^{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
Limit on the left could not be calculated

\lim_{x \to -\infty} \left(-1\right)^{x}

Limit on the right could not be calculated

\lim_{x \to \infty} \left(-1\right)^{x}

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of (-1)^x, divided by x at x->+oo and x ->-oo
Limit on the left could not be calculated

\lim_{x \to -\infty}\left(\frac{\left(-1\right)^{x}}{x}\right)

Limit on the right could not be calculated

\lim_{x \to \infty}\left(\frac{\left(-1\right)^{x}}{x}\right)

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\right)^{x} = \left(-1\right)^{- x}

- No

\left(-1\right)^{x} = - \left(-1\right)^{- x}

- No
so, the function
not is
neither even, nor odd