Here's #1 for you:
Collect the terms on right hand side over a common denominator:
\(\frac{A}{x-2}+\frac{B}{x+1}\rightarrow\frac{A(x+1)+B(x-2)}{(x-2)(x+1)}\rightarrow\frac{(A+B)x+A-2B}{x^2-x-2}\)
Compare this with \(\frac{x+7}{x^2-x-2}\)
To make them the same compare the numerators. It is clear we must have:
A + B = 1 and
A - 2B = 7
Subtract the second from the first to get: 3B = -6 so B = -3
Put this back into the first to get A - 3 = 1 so A = 4
Hence (A, B) = (4, -3)
For #2
Set x = each term in turn and solve for x. if the result lies in the range given for each term then that result is valid. If not it isn't.
For #3
Can't see the whole expression!
For #4
First plug sqrt(2) into h to get, say m = h(sqrt(2))
Then plug m into h to get h(m) which is h(h(sqrt(2)))
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