Explanation:
Two functions of a certain space are orthogonal if its inner product is null. Therefore that two particular functions are orthogonal depends on how the inner product has been defined. The most common definition is as follows:
Two functions f and g, continuous and defined in the interval [a, b] are orthogonal in said interval if:
[tex](f,g)=\int\limits^b_a {f(x})g(x) \, dx =0[/tex]
The functions [tex]f(x)=x^5[/tex] and [tex]g(x)=x^4[/tex] are orthogonal in [-1, 1].
[tex](f,g)=\int\limits^1_{-1} {x^5*x^4} \, dx \\(f,g)=\int\limits^1_{-1} {x^{9}} \, dx\\(f,g)=(\frac{x^{10}}{10})^1_{-1}\\\\(f,g)=\frac{1^{10}}{10}-\frac{-1^{10}}{10}\\\\(f,g)=\frac{1}{10}-\frac{1}{10}=0[/tex]
