Mister Exam

Other calculators

Plotting a function graph/ abs(x^2-2*abs(x)-3)

Graphing y = abs(x^2-2*abs(x)-3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       | 2            |
f(x) = |x  - 2*|x| - 3|
f(x)=∣(x2−2∣x∣)−3∣f{\left(x \right)} = \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| 
f = Abs(x^2 - 2*|x| - 3)
The graph of the function
02468-8-6-4-2-10100100
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:
∣(x2−2∣x∣)−3∣=0\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−3x_{1} = -3
x2=3x_{2} = 3
Numerical solution
x1=3x_{1} = 3
x2=−3x_{2} = -3
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(x^2 - 2*|x| - 3).
∣−3+(02−2∣0∣)∣\left|{-3 + \left(0^{2} - 2 \left|{0}\right|\right)}\right|
The result:
f(0)=3f{\left(0 \right)} = 3
The point:
(0, 3)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(4(x−sign⁡(x))2δ(−x2+2∣x∣+3)+(2δ(x)−1)sign⁡(−x2+2∣x∣+3))=02 \left(4 \left(x - \operatorname{sign}{\left(x \right)}\right)^{2} \delta\left(- x^{2} + 2 \left|{x}\right| + 3\right) + \left(2 \delta\left(x\right) - 1\right) \operatorname{sign}{\left(- x^{2} + 2 \left|{x}\right| + 3 \right)}\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→−∞∣(x2−2∣x∣)−3∣=∞\lim_{x \to -\infty} \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞∣(x2−2∣x∣)−3∣=∞\lim_{x \to \infty} \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 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 Abs(x^2 - 2*|x| - 3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣(x2−2∣x∣)−3∣x)=−∞\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|}{x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(∣(x2−2∣x∣)−3∣x)=∞\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|}{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:
∣(x2−2∣x∣)−3∣=∣(x2−2∣x∣)−3∣\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|
- Yes
∣(x2−2∣x∣)−3∣=−∣(x2−2∣x∣)−3∣\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = - \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|
- No
so, the function
is
even
    © Mister Exam – Calculator