Graphing y = 3^x-3

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        x    
f(x) = 3  - 3

f{\left(x \right)} = 3^{x} - 3

f = 3^x - 3

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:

3^{x} - 3 = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 1

Numerical solution

x_{1} = 1

x_{2} = 1.00000000000008

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to 3^x - 3.

-3 + 3^{0}

The result:

f{\left(0 \right)} = -2

The point:

(0, -2)

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)} =

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)} =

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(3^{x} - 3\right) = -3

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = -3

\lim_{x \to \infty}\left(3^{x} - 3\right) = \infty

Let's take the limit
so,
horizontal asymptote on the right doesn’t exist

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of 3^x - 3, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{3^{x} - 3}{x}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

\lim_{x \to \infty}\left(\frac{3^{x} - 3}{x}\right) = \infty

Let's take the limit
so,
inclined asymptote on the right doesn’t exist

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:

3^{x} - 3 = -3 + 3^{- x}

- No

3^{x} - 3 = 3 - 3^{- x}

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