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Chapter 2: Q24E (page 97)

Which of the functionsf fromR toR in Exercises 21 through 24 are invertible?24 f(x)=x3-x.

Short Answer

Expert verified

The functionfx=x3-x from to is not invertible.

Step by step solution

01

Invertible Function

Functionfx is invertible iff is one-to-one function, that is there are no valuesx1鈮�x2 such that fx1=fx2.If we can find such valuesx1,x2 so thatx1鈮�x2 andfx1=fx2 thenfx is not invertible.

02

Finding such values such that x1≠x2 but f(x1)=f(x2)

It is easy to see thatf-1=f1 but -1鈮�1, sofx is not invertible.

Hence, the given function is not invertible.

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Most popular questions from this chapter

For which values of the constants b and c is the following matrix invertible?

[01b-10-c-b-c0]

Use part (c) of Theorem 2.3.11 to prove part (b): If Ais a regular transition matrix of size n脳nwith equilibrium distribution x鈫�equ, and ifx鈫� is any distribution vector in鈩�n , then limm鈫�鈭�(Amx鈫�)=x鈫�equ.

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation T(x)鈫�=Ax鈫�on this face.

26.[0110]

Question:TRUE OR FALSE?

The matrix product(abcd)d-b-ca is always a scalar multiple of l2.

a. Show that if an invertible n脳nmatrix A admits an LU-factorization, then it admits an LDU factorization. See Exercise 90 d.

b. Show that if an invertible matrix A admits an L DU-factorization, then this factorization is unique. Hint: Suppose that A=L1D1U1=L2D2U2, then U2U-1=D2-1L2-1L1D1is diagonal (why?). Conclude that U2=U1.
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