# Algebra 2 Domain And Range Homework

Note 3: We are talking about the domain and range of functions, which have at most one y-value for each x-value, not relations (which can have more than one.).

## Algebra 2 Domain And Range Homework

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It's always a lot easier to work out the domain and range when reading it off the graph (but we must make sure we zoom in and out of the graph to make sure we see everything we need to see). However, we don't always have access to graphing software, and sketching a graph usually requires knowing about discontinuities and so on first anyway.

Domain and range is heavily emphasized in the Algebra 1 TEKS. The concept is found in two readiness standards and a supporting standard. Having taught it, I thought I would share some tips from my failures and successes.

As teachers, we are all guilty of jumping to an example that students are not ready for, which can cause them to become overwhelmed and shut down. Or worse, they develop false confidence and do an entire set of problems incorrectly. I thought domain and range was really intuitive, so I tried to cover everything in one class period and my students were lost.

I combed through the 2017 and 2018 STAAR tests and out of all the domain and range problems, 80% used a graph. Students have to be able to determine the domain and range by looking at a function on a graph. I recommend having students annotate the graph by use of colored pencils or highlighters. I have seen this done two ways.

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The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates). Only the elements "used" by the relation or function constitute the range.

Topics for 2nd midterm exam: The vector space of linear maps, the rank plus nullity theorem also called the fundamental theorem of linear maps, the matrix of a linear map with respect to bases of the domain and of the range, how the matrix changes when the bases change, proof that the matrix of a composition of linear maps is the product of the matrices of the maps, products and quotients of vector spaces, dual spaces and duals of linear maps, the division algorithm for polynomials, fatorization of polynomials over the real and complex numbers, eigenvalues and eigenvectors, the space of linear operators of a vector space, change of basis and its effect on the matrix of an operator, similar matrices, polynomials of operators, upper triangular matrices, eigenvalues and diagonal matrices

Topics for final exam: All topics for the second midterm, Bases of a vector space and dual bases of the dual space, proof that if M is the matrix of a linear map with respect to bases of its domain and range, then the matrix of the dual map with respect to dual bases is the transpose of M. Column and row rank of a matrix, proof that they are equal, Complex numbers and their absolute value, complex conjugate, Triangle inequality for complex numbers, vector spaces over finite fields, polynomials with coefficients in finite fields, Quotient spaces, Inner product spaces and norms of vectors and metrics on vector spaces, Schwartz inequality and its proof, Orthogonal vectors, orthogonal and orthonormal bases, Parallelogram equality, Gram-Schmidt procedure to get orthonormal sets of vectors, particularly polynomials, Linear operators, If V is a complex vector space and T is a linear operator on V, Find a basis of V that makes the matrix of T upper triangular, Find such an orthonormal basis, The representation theorem for vector spaces with an inner product, Subspaces and orthogonal subspaces, Normal and self-adjoint operators, Hermitian matrices, Proof that for complex vector spaces, normal operators have an orthonormal basis of eigenvectors, Same for self-adjoint operators in real vector spaces, positive operators and positive square roots of operators, Isometries and their properties, null spaces of powers of an operator, Eigenspaces and generalized eigenspaces, nilpotent operators, Proof that every nilpotent operator operator has a strictly upper-triangular matrix with respect to some basis, If T: V --> V is a linear operator (V complex) with eigenvalues a_1, a_2, ..., a_k, prove that V is a direct sum of the generalized eigenspaces, G(a_1, T), G(a_2, T), ..., G(a_k, T). Define the characteristic and minimal polynomials of T, Prove the Hamilton-Jacobi theorem, and prove that the minimal polynomial divides the characteristic polynomial, Define a Jordan basis for an operator T: V --> V.