Problem 3.7(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, withthe same eigenvalue q. Show that any linear combination of f and g is itselfan eigenfunction of Q. with eigenvalue q.(b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of theoperator d?/dx², with the same eigenvalue. Construct two linear combina-tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).

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(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator 2, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q, with eigenvalue q.