### Basic functions: Power functions

### Sums of power functions

We look at how power functions of the form \[f(x)=a\cdot x^p+b\cdot x^q, \quad \text{met }p\neq q\] behave for \(x\) near zero or for large values of \(x\).

The diagrams below show the graphs of \(x^2-4x^{\frac{1}{2}}\), \(x^2\) and \(4x^{\frac{1}{2}}\); on the left-hand side for the interval \([0,8]\) and on the right-hand side for the interval \([40,48]\).

By comparing the graphs we see that the graph of \(f(x)=x^2-4x^{\frac{1}{2}}\) for \(x\) near \(0\) resembles the graph of \(-4x^{\frac{1}{2}}\) and looks more like the graph of \(x^2\) for large values of \(x\) .

In other words, the \(f(x)=x^2-4x^{\frac{1}{2}}\) function resembles \(x\mapsto -4x^{\frac{1}{2}}\) for \(x\) near \(0\) and is more similar to the function \(x\mapsto x^2\) \(x\) for large values of \(x\).

This can also be seen by bringing out one of the powers as a factor.

It follows from \(f(x)=x^2-4x^{\frac{1}{2}}=-4x^{\frac{1}{2}}\cdot \left(1-\tfrac{1}{4}x^{\frac{3}{2}}\right)\) that for small values of \(x\) the term \(x^{\frac{3}{2}}\) is very small (at least much smaller than \(x^{\frac{1}{2}}\) ) and the expression between brackets is close to \(1\). Then \(f(x)\) looks like \(x\mapsto -4x^{\frac{1}{2}}\).

It follows from \(f(x)=x^2-4x^{\frac{1}{2}}=x^2\cdot \left(1-4x^{-\frac{3}{2}}\right)\) that for large values of \(x\) the term \(x^{-\frac{3}{2}}\) is very small and the expression between brackets is close to \(1\). Then \(f(x)\) looks like \(x\mapsto x^2\).

In a sum of two or more power functions, the function resembles for large values of \(x\) the power term with the largest exponent and resembles the power term with the smallest exponent for small values of \(x\).