What is the domain of the function y sqrt X?

That's absolutely correct. The domain of a function is the set of all input values that you're "legally" allowed to plug into the function. For the function $y=\sqrt {x-2}$, that's going to be $x\geq 2$ because if you were to plug in, say, -1, you would end up with a negative number under the square root and, as you probably know, the square root function is not defined for negatives in the real number system.

The range of a function is usually a bit tougher to find. In your case, the range is the same as that of $f(x)=\sqrt{x}$: $[0,\infty)$. How do we know that? Well, because the square root function is one of those well-known functions whose behavior we all should be familiar with. And we also know, that when you add or subtract a constant before the prevailing operation takes place (for $2(x+1)^2-1$, the prevailing operation would be squaring, for $5\sqrt{x+2}+1$—taking the square root), you are shifting the graph left or right. So, in our case here, the graph is shifted 2 units to the right. And that's the only transformation that's happening. No shifts up or down. So, the range does not change.

The domain of the function will be determined by the fact that the expression that's under the radical must be positive for real numbers.

Since#x^2#will always be positive regardless of the sign of#x#, you need to find the values of#x#that will make#x^2#smaller than#1#, since those are the only values that will make the expression negative.

So, you need to have

#x^2 - 1 >=0#

#x^2 >=1#

Take the square root of both sides to get

#|x| >= 1#

This of course means that you have

#x >= 1" "#and#" "x<=-1#

The domain of the function will thus be#(-oo, -1] uu [1, + oo)#.

The range of the function will be determined by the fact that the square root of a real number must always be positive. The smallest value the function can take will happen for#x = -1#and for#x=1#, since those values of#x#will make the radical term equal to zero.

For interp2, scattered points consist of a pair of arrays that define a collection of points scattered in 2-D space. One array contains the x-coordinates, and the other contains the y-coordinates.

For example, the following code specifies the points, (2,7), (5,3), (4,1), and (10,9):

x = [2 5; 4 10]; y = [7 3; 1 9];

Mathematical function returning -1, 0 or 1

Signum function y=sgn⁡x{\displaystyle y=\operatorname {sgn} x}

In mathematics, the sign function or signum function (from , Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as sgn⁡(x){\displaystyle \operatorname {sgn}(x)}. To avoid confusion with the sine function, this function is usually called the signum function.

Definition[edit]

The signum function of a real number x{\displaystyle x} is a piecewise function which is defined as follows:

sgn⁡x:={−1if x<0,0if x=0,1if x>0.{\displaystyle \operatorname {sgn} x:={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}

Properties[edit]

The sign function is not continuous at x=0{\displaystyle x=0}.

Any real number can be expressed as the product of its absolute value and its sign function:

x=|x|sgn⁡x.{\displaystyle x=|x|\operatorname {sgn} x.}

It follows that whenever x{\displaystyle x} is not equal to 0 we have

sgn⁡x=x|x|=|x|x.{\displaystyle \operatorname {sgn} x={\frac {x}{|x|}}={\frac {|x|}{x}}\,.}

Similarly, for any real number x{\displaystyle x},

|x|=xsgn⁡x.{\displaystyle |x|=x\operatorname {sgn} x.}

We can also ascertain that:

sgn⁡xn=(sgn⁡x)n.{\displaystyle \operatorname {sgn} x^{n}=(\operatorname {sgn} x)^{n}.}

The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval [−1,1]{\displaystyle [-1,1]}, "filling in" the sign function (the subdifferential of the absolute value is not single-valued at 0). Note, the resultant power of x{\displaystyle x} is 0, similar to the ordinary derivative of x{\displaystyle x}. The numbers cancel and all we are left with is the sign of x{\displaystyle x}.

d|x|dx=sgn⁡x for x≠0.{\displaystyle {\frac {d|x|}{dx}}=\operatorname {sgn} x{\text{ for }}x\neq 0\,.}

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function, which can be demonstrated using the identity

sgn⁡x=2H(x)−1,{\displaystyle \operatorname {sgn} x=2H(x)-1\,,}

where H(x){\displaystyle H(x)} is the Heaviside step function using the standard H(0)=12{\displaystyle H(0)={\frac {1}{2}}} formalism. Using this identity, it is easy to derive the distributional derivative:[3]

dsgn⁡xdx=2dH(x)dx=2δ(x).{\displaystyle {\frac {d\operatorname {sgn} x}{dx}}=2{\frac {dH(x)}{dx}}=2\delta (x)\,.}

The Fourier transform of the signum function is

∫−∞∞(sgn⁡x)e−ikxdx=p.v.2ik,{\displaystyle \int _{-\infty }^{\infty }(\operatorname {sgn} x)e^{-ikx}dx=\mathrm {p.v.} {\frac {2}{ik}},}

where p. v.{\displaystyle {\text{p. v.}}} means Cauchy principal value.

The signum can also be written using the Iverson bracket notation:

sgn⁡x=−[x<0]+[x>0].{\displaystyle \operatorname {sgn} x=-[x<0]+[x>0]\,.}

The signum can also be written using the floor and the absolute value functions:

sgn⁡x=⌊x|x|+1⌋−⌊−x|x|+1⌋.{\displaystyle \operatorname {sgn} x={\Biggl \lfloor }{\frac {x}{|x|+1}}{\Biggr \rfloor }-{\Biggl \lfloor }{\frac {-x}{|x|+1}}{\Biggr \rfloor }\,.}

The signum function has a very simple definition if 00{\displaystyle 0^{0}} is accepted to be equal to 1. Then signum can be written for all real numbers as

sgn⁡x=0(−x+|x|)−0(x+|x|).{\displaystyle \operatorname {sgn} x=0^{\left(-x+\left\vert x\right\vert \right)}-0^{\left(x+\left\vert x\right\vert \right)}\,.}

The signum function coincides with the limits

sgn⁡x=limn→∞1−2−nx1+2−nx.{\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }{\frac {1-2^{-nx}}{1+2^{-nx}}}\,.}

and

sgn⁡x=limn→∞2πtan−1⁡(nx).{\displaystyle \operatorname {sgn} x=\lim _{n\to \infty }{\frac {2}{\pi }}\tan ^{-1}(nx)\,.}

For k>1{\displaystyle k>1}, a smooth approximation of the sign function is

sgn⁡x≈tanh⁡kx.{\displaystyle \operatorname {sgn} x\approx \tanh kx\,.}

Another approximation is

sgn⁡x≈xx2+ε2.{\displaystyle \operatorname {sgn} x\approx {\frac {x}{\sqrt {x^{2}+\varepsilon ^{2}}}}\,.}

which gets sharper as ε→0{\displaystyle \varepsilon \to 0}; note that this is the derivative of x2+ε2{\displaystyle {\sqrt {x^{2}+\varepsilon ^{2}}}}. This is inspired from the fact that the above is exactly equal for all nonzero x{\displaystyle x} if ε=0{\displaystyle \varepsilon =0}, and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of x2+y2{\displaystyle {\sqrt {x^{2}+y^{2}}}}).

See .

Complex signum[edit]

The signum function can be generalized to complex numbers as:

sgn⁡z=z|z|{\displaystyle \operatorname {sgn} z={\frac {z}{|z|}}}

for any complex number z{\displaystyle z} except z=0{\displaystyle z=0}. The signum of a given complex number z{\displaystyle z} is the point on the unit circle of the complex plane that is nearest to z{\displaystyle z}. Then, for z≠0{\displaystyle z\neq 0},

sgn⁡z=eiarg⁡z,{\displaystyle \operatorname {sgn} z=e^{i\arg z}\,,}

where arg{\displaystyle \arg } is the complex argument function.

For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines, for z=0{\displaystyle z=0}:

sgn⁡(0+0i)=0{\displaystyle \operatorname {sgn}(0+0i)=0}

Another generalization of the sign function for real and complex expressions is csgn{\displaystyle {\text{csgn}}}, which is defined as:

csgn⁡z={1if Re(z)>0,−1if Re(z)<0,sgn⁡Im(z)if Re(z)=0{\displaystyle \operatorname {csgn} z={\begin{cases}1&{\text{if }}\mathrm {Re} (z)>0,\\-1&{\text{if }}\mathrm {Re} (z)<0,\\\operatorname {sgn} \mathrm {Im} (z)&{\text{if }}\mathrm {Re} (z)=0\end{cases}}}

where Re(z){\displaystyle {\text{Re}}(z)} is the real part of z{\displaystyle z} and Im(z){\displaystyle {\text{Im}}(z)} is the imaginary part of z{\displaystyle z}.

We then have (for z≠0{\displaystyle z\neq 0}):

csgn⁡z=zz2=z2z.{\displaystyle \operatorname {csgn} z={\frac {z}{\sqrt {z^{2}}}}={\frac {\sqrt {z^{2}}}{z}}.}

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