Mister Exam
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How to use it?
- Graphing y =:
-
sqrt((1-(abs(x)-1)^2))
-
3x^4-4x^3-12x^2+3
-
x^3-5x^2+7x-3
-
e^-x
-
Identical expressions
- sqrt((one -(abs(x)- one)^ two))
- square root of ((1 minus (abs(x) minus 1) squared ))
- square root of ((one minus (abs(x) minus one) to the power of two))
- √((1-(abs(x)-1)^2))
- sqrt((1-(abs(x)-1)2))
- sqrt1-absx-12
- sqrt((1-(abs(x)-1)²))
- sqrt((1-(abs(x)-1) to the power of 2))
- sqrt1-absx-1^2
-
Similar expressions
- sqrt((1+(abs(x)-1)^2))
- sqrt((1-(abs(x)+1)^2))
Graphing y = sqrt((1-(abs(x)-1)^2))
The solution
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f(x) = \/ 1 - (|x| - 1)
$$f{\left(x \right)} = \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}$$
f = sqrt(1 - (|x| - 1*1)^2)
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:
$$\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
Numerical solution
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
so we need to solve the equation:
$$\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
Numerical solution
$$x_{1} = -2$$
$$x_{2} = 0$$
$$x_{3} = 2$$
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:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{\left(\left|{x}\right| - 1\right) \operatorname{sign}{\left(x \right)}}{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}} = 0$$
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
$$- \frac{\left(\left|{x}\right| - 1\right) \operatorname{sign}{\left(x \right)}}{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}} = 0$$
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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- \frac{\frac{\left(\left|{x}\right| - 1\right)^{2} \operatorname{sign}^{2}{\left(x \right)}}{- \left(\left|{x}\right| - 1\right)^{2} + 1} + 2 \left(\left|{x}\right| - 1\right) \delta\left(x\right) + \operatorname{sign}^{2}{\left(x \right)}}{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}} = 0$$
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
$$- \frac{\frac{\left(\left|{x}\right| - 1\right)^{2} \operatorname{sign}^{2}{\left(x \right)}}{- \left(\left|{x}\right| - 1\right)^{2} + 1} + 2 \left(\left|{x}\right| - 1\right) \delta\left(x\right) + \operatorname{sign}^{2}{\left(x \right)}}{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}} = 0$$
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
$$\lim_{x \to -\infty} \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = \infty i$$
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 sqrt(1 - (|x| - 1*1)^2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}}{x}\right) = - i$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - i x$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}}{x}\right) = i$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = i x$$
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}}{x}\right) = - i$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - i x$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}}{x}\right) = i$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = i x$$
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:
$$\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}$$
- Yes
$$\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = - \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}$$
- No
so, the function
is
even
So, check:
$$\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}$$
- Yes
$$\sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1} = - \sqrt{- \left(\left|{x}\right| - 1\right)^{2} + 1}$$
- No
so, the function
is
even
The graph