A1 - Linear Polynomial Transformations


Example 1

A1 - Linear Polynomial Transformations (ver. 1)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-4 \, h\left(x\right)^{2} - 2 \, h'\left(x\right)\hspace{3em}T(h(x))=2 \, h\left(-5\right) + 2 \, h\left(x^{2}\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 2

A1 - Linear Polynomial Transformations (ver. 2)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=5 \, x h\left(x\right) + h'\left(x\right)\hspace{3em}T(h(x))=-h\left(x\right) h'\left(x\right) - 5 \, h\left(-2\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 3

A1 - Linear Polynomial Transformations (ver. 3)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-5 \, g\left(x\right)^{3} - 4 \, g\left(-4\right)\hspace{3em}T(g(x))=-4 \, g\left(x^{3}\right) - g'\left(4\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 4

A1 - Linear Polynomial Transformations (ver. 4)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=2 \, g\left(x\right) g'\left(x\right) - 5 \, g'\left(-3\right)\hspace{3em}T(g(x))=-3 \, g\left(5\right) - 2 \, g'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 5

A1 - Linear Polynomial Transformations (ver. 5)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=4 \, x^{2} f\left(x\right) - 4 \, f'\left(x\right)\hspace{3em}T(f(x))=-5 \, f\left(-3\right) + 1\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 6

A1 - Linear Polynomial Transformations (ver. 6)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, h\left(x\right)^{2} - 3 \, h\left(x^{2}\right)\hspace{3em}T(h(x))=5 \, h\left(-4\right) + 5 \, h'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 7

A1 - Linear Polynomial Transformations (ver. 7)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=5 \, x^{3} h\left(x\right) + h\left(x\right)^{2}\hspace{3em}T(h(x))=-2 \, h\left(x^{2}\right) + h'\left(4\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 8

A1 - Linear Polynomial Transformations (ver. 8)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, h\left(x\right) h'\left(x\right) - 4 \, h\left(-5\right)\hspace{3em}T(h(x))=3 \, h\left(x^{2}\right) + h'\left(3\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 9

A1 - Linear Polynomial Transformations (ver. 9)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=3 \, g\left(x\right)^{3} + g\left(x^{2}\right)\hspace{3em}T(g(x))=4 \, x g\left(x\right) + 5 \, g\left(-2\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 10

A1 - Linear Polynomial Transformations (ver. 10)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=5 \, x h\left(x\right) + 5 \, h'\left(x\right)\hspace{3em}T(h(x))=h\left(x\right)^{3} + 5 \, h\left(x^{2}\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 11

A1 - Linear Polynomial Transformations (ver. 11)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-4 \, g'\left(-2\right) + 2 \, g'\left(x\right)\hspace{3em}T(g(x))=x g\left(x\right) - 4\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 12

A1 - Linear Polynomial Transformations (ver. 12)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-3 \, g\left(x\right) g'\left(x\right) - 3 \, g'\left(x\right)\hspace{3em}T(g(x))=4 \, x^{2} g\left(x\right) + 5 \, g\left(x^{2}\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 13

A1 - Linear Polynomial Transformations (ver. 13)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-4 \, f\left(x\right)^{3} + f\left(x\right)\hspace{3em}T(f(x))=x^{3} f\left(x\right) - f'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 14

A1 - Linear Polynomial Transformations (ver. 14)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-2 \, f\left(x\right) f'\left(x\right) + 4 \, f\left(x\right)\hspace{3em}T(f(x))=5 \, f\left(x^{3}\right) + 4 \, f'\left(2\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 15

A1 - Linear Polynomial Transformations (ver. 15)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, x^{2} h\left(x\right) - 4 \, x\hspace{3em}T(h(x))=-3 \, h'\left(-4\right) - 5 \, h'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 16

A1 - Linear Polynomial Transformations (ver. 16)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=3 \, g\left(x\right) + 2 \, g'\left(x\right)\hspace{3em}T(g(x))=-4 \, g\left(x\right) g'\left(x\right) + 2 \, g'\left(1\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 17

A1 - Linear Polynomial Transformations (ver. 17)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=g\left(3\right) + 4 \, g\left(x^{2}\right)\hspace{3em}T(g(x))=-g\left(x\right) g'\left(x\right) + g'\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 18

A1 - Linear Polynomial Transformations (ver. 18)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, h\left(x^{2}\right) + 4 \, h'\left(x\right)\hspace{3em}T(h(x))=-5 \, x - 3 \, h\left(-5\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 19

A1 - Linear Polynomial Transformations (ver. 19)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, x^{3} g\left(x\right) + g\left(2\right)\hspace{3em}T(g(x))=5 \, x^{2} - g'\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 20

A1 - Linear Polynomial Transformations (ver. 20)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=2 \, x^{2} g\left(x\right) - g\left(x^{2}\right)\hspace{3em}T(g(x))=-3 \, g\left(x\right) g'\left(x\right) + g'\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 21

A1 - Linear Polynomial Transformations (ver. 21)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-4 \, f\left(x\right) f'\left(x\right) - 3 \, f\left(x\right)\hspace{3em}T(f(x))=-f'\left(-4\right) + f'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 22

A1 - Linear Polynomial Transformations (ver. 22)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=4 \, x^{3} - 2 \, f'\left(3\right)\hspace{3em}T(f(x))=2 \, x f\left(x\right) + 2 \, f\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 23

A1 - Linear Polynomial Transformations (ver. 23)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-f\left(x\right)^{2} - 5 \, f'\left(x\right)\hspace{3em}T(f(x))=-4 \, f\left(x\right) - 5 \, f'\left(-3\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 24

A1 - Linear Polynomial Transformations (ver. 24)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(x\right)^{3} - 2 \, g'\left(-2\right)\hspace{3em}T(g(x))=3 \, g\left(x\right) - 2 \, g'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 25

A1 - Linear Polynomial Transformations (ver. 25)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(2\right) - 3 \, g'\left(x\right)\hspace{3em}T(g(x))=-4 \, g\left(x\right) g'\left(x\right) - 3 \, g'\left(2\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 26

A1 - Linear Polynomial Transformations (ver. 26)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, x^{2} h\left(x\right) + 2 \, h\left(3\right)\hspace{3em}T(h(x))=4 \, x^{3} + 4 \, h'\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 27

A1 - Linear Polynomial Transformations (ver. 27)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-5 \, x^{3} f\left(x\right) - 3 \, f'\left(x\right)\hspace{3em}T(f(x))=3 \, f\left(x\right)^{2} - 3 \, f'\left(1\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 28

A1 - Linear Polynomial Transformations (ver. 28)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-2 \, x^{3} g\left(x\right) + 4 \, g\left(x\right)^{2}\hspace{3em}T(g(x))=-2 \, g'\left(4\right) + 5 \, g'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 29

A1 - Linear Polynomial Transformations (ver. 29)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-2 \, h\left(x\right)^{3} - 3 \, h\left(-3\right)\hspace{3em}T(h(x))=-3 \, h\left(x\right) - 5 \, h'\left(2\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 30

A1 - Linear Polynomial Transformations (ver. 30)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-2 \, x^{3} - 5 \, g'\left(-4\right)\hspace{3em}T(g(x))=5 \, g\left(x\right) - g'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 31

A1 - Linear Polynomial Transformations (ver. 31)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-2 \, f\left(-5\right) - 2 \, f'\left(x\right)\hspace{3em}T(f(x))=2 \, x^{2} f\left(x\right) - 2 \, f\left(x\right)^{3}\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 32

A1 - Linear Polynomial Transformations (ver. 32)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-5 \, g\left(x\right) g'\left(x\right) + g\left(x^{3}\right)\hspace{3em}T(g(x))=-2 \, g\left(2\right) + 3 \, g'\left(4\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 33

A1 - Linear Polynomial Transformations (ver. 33)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-5 \, f\left(1\right) + 5 \, f\left(x^{3}\right)\hspace{3em}T(f(x))=4 \, x f\left(x\right) + f\left(x\right) f'\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 34

A1 - Linear Polynomial Transformations (ver. 34)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-3 \, g\left(x\right) g'\left(x\right) - 2 \, g\left(x\right)\hspace{3em}T(g(x))=-3 \, x^{3} g\left(x\right) - g'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 35

A1 - Linear Polynomial Transformations (ver. 35)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-x^{3} - 2 \, f'\left(x\right)\hspace{3em}T(f(x))=5 \, x^{3} f\left(x\right) + 3 \, f\left(x^{2}\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 36

A1 - Linear Polynomial Transformations (ver. 36)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-h\left(x\right)^{3} + h\left(-2\right)\hspace{3em}T(h(x))=4 \, h\left(x^{2}\right) - 3 \, h'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 37

A1 - Linear Polynomial Transformations (ver. 37)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=2 \, x^{2} + 4 \, h'\left(1\right)\hspace{3em}T(h(x))=5 \, x^{2} h\left(x\right) + 4 \, h\left(-2\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 38

A1 - Linear Polynomial Transformations (ver. 38)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=5 \, f\left(1\right) - f'\left(x\right)\hspace{3em}T(f(x))=-3 \, f\left(x\right)^{3} + 2 \, f\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 39

A1 - Linear Polynomial Transformations (ver. 39)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-2 \, h\left(-1\right) - 2 \, h'\left(x\right)\hspace{3em}T(h(x))=2 \, h\left(x\right) h'\left(x\right) - 5 \, h'\left(-5\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 40

A1 - Linear Polynomial Transformations (ver. 40)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(x\right)^{3} + 3 \, g'\left(x\right)\hspace{3em}T(g(x))=2 \, x g\left(x\right) - 5 \, g'\left(1\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 41

A1 - Linear Polynomial Transformations (ver. 41)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=x^{3} f\left(x\right) + 3 \, f'\left(x\right)\hspace{3em}T(f(x))=-4 \, x^{2} - 4 \, f\left(x^{3}\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 42

A1 - Linear Polynomial Transformations (ver. 42)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=-2 \, g'\left(-1\right) - 3 \, g'\left(x\right)\hspace{3em}T(g(x))=2 \, x^{3} + 5 \, g\left(2\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 43

A1 - Linear Polynomial Transformations (ver. 43)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=-f\left(x\right) f'\left(x\right) - 2 \, f'\left(x\right)\hspace{3em}T(f(x))=4 \, f\left(3\right) + 5 \, f\left(x^{2}\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 44

A1 - Linear Polynomial Transformations (ver. 44)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=2 \, x + 5 \, f\left(x\right)\hspace{3em}T(f(x))=-3 \, f\left(-3\right) + 5 \, f'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 45

A1 - Linear Polynomial Transformations (ver. 45)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-x^{2} h\left(x\right) + 4 \, h'\left(-2\right)\hspace{3em}T(h(x))=5 \, h\left(x\right) h'\left(x\right) + h\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 46

A1 - Linear Polynomial Transformations (ver. 46)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=4 \, g\left(x\right)^{3} + 5 \, g\left(-5\right)\hspace{3em}T(g(x))=2 \, x^{3} g\left(x\right) + 2 \, g'\left(x\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 47

A1 - Linear Polynomial Transformations (ver. 47)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(h(x))=-h\left(x\right) h'\left(x\right) + 4 \, h'\left(x\right)\hspace{3em}T(h(x))=5 \, x h\left(x\right) + 5 \, h\left(-1\right)\]

Answer.

\(S\) is non-linear, and \(T\) is linear.


Example 48

A1 - Linear Polynomial Transformations (ver. 48)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(f(x))=f\left(4\right) + f'\left(x\right)\hspace{3em}T(f(x))=5 \, f\left(x\right) f'\left(x\right) + 4 \, f'\left(1\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 49

A1 - Linear Polynomial Transformations (ver. 49)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=5 \, g\left(-1\right) - 5 \, g\left(x\right)\hspace{3em}T(g(x))=x - 2 \, g'\left(4\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.


Example 50

A1 - Linear Polynomial Transformations (ver. 50)

Explain why one of the following polynomial transformations is linear, and why the other is not.

\[S(g(x))=g\left(3\right) + 2 \, g\left(x^{3}\right)\hspace{3em}T(g(x))=-4 \, x^{2} - g\left(x\right)\]

Answer.

\(S\) is linear, and \(T\) is non-linear.