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In this example, why is the range given as $Range(L)= (1+x, x)$ and not as simply $(1, x)$.

I thought that it would be $(1,x)$ since we can use a linear combination of $1$ and $x$ to express the form $(a+b) + (a+b+c)x$.

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Is $P_1$ the set of polynomials of degree $\leq 1$?

In any case, the span of $(1,x)$ and $(1+x,x)$ is the same. That is to say, any polynomial that you can write as a linear combination of $1$ and $x$ you can also write as a linear combination of $1+x$ and $x$ and vice versa.

For example, we can write the polynomial $a+bx$ as $a(1+x) + (b-a)x$.

Also, as a side note, it would be better if you used mathjax to type the content from the image, or at least properly cropped the image.

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