How do I norm $5x+4$?

Question

So I would just simply write that like $\frac{5x+4}{\sqrt{41}}$. But solution says that correct answer is $\frac{5x+4}{146}*\sqrt{438}$. How do they get that?

Answer 1

How do they get that?

I'm guessing using the inner product

$$\langle g , h \rangle = \int_{-1}^{1} g(x)h(x) \> dx$$

We see that

$$\|5x+4\| = \left(\int_{-1}^{1} (5x+4)^2 \> dx\right)^{1/2} = \sqrt{\frac{146}{3}}$$

and therefore we normalize with

$$\sqrt{\frac{3}{146}} = \frac{\sqrt{438}}{146}$$

giving us the result your solution guide gives.