Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- -x^3+3x^2-2
- x^3-12*x+2
- 3x^2-2x
-
|x|x+3|x|-5x
-
Identical expressions
- log(one / two)(x+ one)
- logarithm of (1 divide by 2)(x plus 1)
- logarithm of (one divide by two)(x plus one)
- log1/2x+1
- log(1 divide by 2)(x+1)
-
Similar expressions
- log(1/2)(x-1)
- log(1/2)*(x+1)-2
Graphing y = log(1/2)(x+1)
The solution
You have entered
[src]
f(x) = log(1/2)*(x + 1)
$$f{\left(x \right)} = \left(x + 1\right) \log{\left(\frac{1}{2} \right)}$$
f = (x + 1)*log(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:
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
Numerical solution
$$x_{1} = -1$$
so we need to solve the equation:
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
Numerical solution
$$x_{1} = -1$$
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:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\log{\left(\frac{1}{2} \right)} = 0$$
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
$$\log{\left(\frac{1}{2} \right)} = 0$$
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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$0 = 0$$
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
$$0 = 0$$
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
$$\lim_{x \to -\infty}\left(\left(x + 1\right) \log{\left(\frac{1}{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x + 1\right) \log{\left(\frac{1}{2} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(x + 1\right) \log{\left(\frac{1}{2} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x + 1\right) \log{\left(\frac{1}{2} \right)}\right) = -\infty$$
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 log(1/2)*(x + 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \log{\left(\frac{1}{2} \right)}}{x}\right) = - \log{\left(2 \right)}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x \log{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\frac{1}{2} \right)}}{x}\right) = - \log{\left(2 \right)}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x \log{\left(2 \right)}$$
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \log{\left(\frac{1}{2} \right)}}{x}\right) = - \log{\left(2 \right)}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x \log{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \log{\left(\frac{1}{2} \right)}}{x}\right) = - \log{\left(2 \right)}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x \log{\left(2 \right)}$$
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:
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = \left(1 - x\right) \log{\left(\frac{1}{2} \right)}$$
- No
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = - \left(1 - x\right) \log{\left(\frac{1}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = \left(1 - x\right) \log{\left(\frac{1}{2} \right)}$$
- No
$$\left(x + 1\right) \log{\left(\frac{1}{2} \right)} = - \left(1 - x\right) \log{\left(\frac{1}{2} \right)}$$
- No
so, the function
not is
neither even, nor odd