Homogeneous Function Definition in math with Examples

What is Homogeneous Function Definition:

A function $f$ defined by

$u=f(x,y,z,...)$

of any number of variables are said to be homogeneous of degree $n$ in these variables if multiplication of these variables by any number $t\neq0$ result in the multiplication of the function by $t^n \;\;,i.e,$ .

$f(tx,ty,tz,...)= t^n f(x,y,z,...)\;\;\;\;\;\;\;\;\;(i)$

provide that $(tx,ty,tz,...)$ is in the domain of $f$ .

Taking $t=\frac 1x ,\;\; x\neq0$ the equation $(i)$ becomes

$f\left(1,\;\frac yx,\frac zx,…\right)\;=\frac1{x^n}f\left(x,y,z,…\right)$ .

$or\;\;\;f\left(x,y,z,…\right)=x^nf\left(1,\;\frac yx,\frac zx,…\right)$ .

$\Rightarrow\;f\left(x,y,z,…\right)=x^ng\left(\frac yx,\frac zx,…\right)$ .

which is another criterion for a function to be in a homogeneous function. so after the homogeneous function definition now come to the examples

Homogeneous Function Example: 1

consider the function $f$ defined by

$f\left(x,y\right)=\frac xy+\frac34\frac yx+\cos\sqrt{\frac yx}+\ln\left(x\right)-\ln\left(y\right)$ .

$f\left(tx,ty\right)=\frac{tx}{ty}+\frac34\frac{ty}{tx}+\cos\sqrt{\frac{ty}{tx}}+\ln\left(tx\right)-\ln\left(ty\right)$ .

$f\left(tx,ty\right)=\frac{tx}{ty}+\frac34\frac{ty}{tx}+\cos\sqrt{\frac{ty}{tx}}+\ln\;t+\ln\;x-\ln t\;-\ln\;y$ .

$\Rightarrow f\left(tx,ty\right)=\frac xy+\frac34\frac yx+\cos\sqrt{\frac yx}+\ln\;x\;-\ln\;y$ .

$\Rightarrow f\left(tx,ty\right)=f\left(x,y\right)=t^0f\left(x,y\right)$ .

Thus f is homogenous function of degree 0.

Homogeneous Function Example: 2

Let

$f(x,y)=\frac{\sqrt y+\sqrt x}{y+x}$ .

Here

$f(x,y)=\frac{\sqrt y+\sqrt x}{y+x}=\frac{\sqrt x\left[1+\sqrt{\displaystyle\frac yx}\;\right]}{x\left[1+{\displaystyle\frac yx}\right]}$ .

$f(x,y)=x^\frac{-1}2\frac{\left[1+\sqrt{\displaystyle\frac yx}\;\right]}{\left[1+{\displaystyle\frac yx}\right]}$ .

Thus $f(x,y)$ is a homogeneous function of degree $\frac{-1}2$ .

Another Method:

Let

$f(x,y)=\frac{\sqrt y+\sqrt x}{y+x}$ .

Then

$f(tx,ty)=\frac{\sqrt{ty}+\sqrt{tx}}{ty+tx}$ .

$f(tx,ty)=\frac{t^{\displaystyle\frac12}\left(\sqrt y+\sqrt x\right)}{t\;\left(y+x\right)}$

$f(tx,ty)=\frac{t^{\displaystyle\frac{-1}2}\left(\sqrt y+\sqrt x\right)}{\;\left(y+x\right)}$ .

$f(tx,ty)=t^\frac{-1}2f(x,y)$ .

Thus $f(x,y)$ is a homogeneous function of degree $\frac{-1}2$ .

also check homogeneous equation you can also see this topic from here

Homogeneous Function Definition, Examples

Homogeneous Function Example: 1

Homogeneous Function Example: 2

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