An \(m\)-by-\(n\) matrix \(A\) is called totally positive (nonnegative) if every minor of \(A\) is positive (nonnegative). A matrix polynomial function is defined as a matrix whose entries are polynomials in a single variable, e. g., \(A(x)=(a_{ij}(x))\), in which each \(a_{ij}(x)\) is an independent polynomial function. A matrix polynomial function is totaly positive (TP) or totally nonnegative (TN) if \(A(x)\) is TP (or TN) when evaluated at each value of the real variable \(x\). It is easy to show that between two rows (or two columns) of an \(m\)-by-\(n\) TN matrix \(A\) a new polynomial row (or column) may be inserted to form an \((m+1)\)-by-\(n\) (or \(m\)-by-\((n+1)\)) TN matrix \(\hat{A}\). However, if TN is replaced by TP, solvability of the line insertion problem is less clear. Here, the line insertion problem is discussed and generalized to TP matrix functions. It is shown that the line insertion problem always has a solution for TP matrix polynomials, TP matrix rational functions, TP matrix continuous functions and TP matrix differentiable functions.