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Question: this is quantum chemistry please explain and show work i...

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This is quantum chemistry. Please explain and show work. I know the answer but I don’t understand.

Which of the following operators are linear operators? a. The absolute value operator, A, which takes the absolute value of the real and imaginary parts of a function, so that if a function f (x), is given by g(x) + ih(x), where g(x) and h(x) are real functions, then Af (xA(gx) ihx)) Igx)ilh(x)l.

d. The operator, I, which multiplies the function it acts on by x, and then integrates from x-0 to x, so that Î f (x) = x2 f(x)dx

Now I(J(x)+g(x))-x(J(x)+g(x))dx And l(af x))-ax\f (x))dx, where a is any constant ai(() Therefore I is alinear operator Why is there x in the integral instead of x^2 for part d? Why even have two dummy functions? ax where a is the constant ? What about the x?
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