Graphing y = |2x-3|+x+4

You have entered [src]

f(x) = |2*x - 3| + x + 4

f{\left(x \right)} = \left(x + \left|{2 x - 3}\right|\right) + 4

f = x + |2*x - 3| + 4

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(x + \left|{2 x - 3}\right|\right) + 4 = 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 |2*x - 3| + x + 4.

\left|{-3 + 0 \cdot 2}\right| + 4

The result:

f{\left(0 \right)} = 7

The point:

(0, 7)

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)} =

2 \operatorname{sign}{\left(2 x - 3 \right)} + 1 = 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)} =

8 \delta\left(2 x - 3\right) = 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(x + \left|{2 x - 3}\right|\right) + 4\right) = \infty

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

\lim_{x \to \infty}\left(\left(x + \left|{2 x - 3}\right|\right) + 4\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 |2*x - 3| + x + 4, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left(x + \left|{2 x - 3}\right|\right) + 4}{x}\right) = -1

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

y = - x

\lim_{x \to \infty}\left(\frac{\left(x + \left|{2 x - 3}\right|\right) + 4}{x}\right) = 3

Let's take the limit
so,
inclined asymptote equation on the right:

y = 3 x

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(x + \left|{2 x - 3}\right|\right) + 4 = - x + \left|{2 x + 3}\right| + 4

- No

\left(x + \left|{2 x - 3}\right|\right) + 4 = x - \left|{2 x + 3}\right| - 4

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