Mathematics for Engineering I

Brandenburg Technical University (BTU) Cottbus

Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm

Mathematics for Engineering I

Problem Sheet No. 3, November 5/6, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Given the function

f(x) = sin(x), sketch the graphs of the following functions:

f₁(x) = f(x) + 1, f₂(x) =f(x+ 1), f₃(x) = 2f(x), f₄(x) = f(2x) f5(x) = −f(x), f6(x) =f(−x), f7(x) = 1

f(x) , f8(x) = f

1

x

f9(x) = f(x)² , f10(x) =f(x²).

2. Find the maximal domain of definition and the range of the following functions:

a) f(x) = x

x+ 1 b) f(x) = x³ x²−1

3. Check if the following functions are injective and/or surjective. Which of them are bijective? If possible, give the inverse function.

f1 : [0,∞)−→R, f1(x) =

√x−2

√x+ 1 , f₂ : [0,∞)−→[4,∞), f₂(x) = (x+ 2)² , f₃ : [4,∞)−→[0,1), f₃(x) =

√x−2

√x+ 1 ,

4. Transform the following terms to make the denominator free of any roots, e. g.

√1 2 =

√2 2 .

Write the numerator in the simplest form you can find.

2 +√

√ 2

2 , 1−√

2 1 +√

2

a+b

√a−√

b , (a, b≥0, a6=b)

Homework

1. Given the following function:

a) f(x) = x²+ 1, b) f(x) = x+ 1

x , (x6= 0) . Sketch the graphs and compute the formulas of the following functions:

f₁(x) = f(x) + 1, f₂(x) =f(x+ 1), f₃(x) = 2f(x), f₄(x) = f(2x) f₅(x) = −f(x), f₆(x) =f(−x), f₇(x) = 1

f(x) , f₈(x) = f1 x

f₉(x) = f(x)² , f₁₀(x) =f(x²).

2. Find the maximal domain of definition and the range (image) of the following functions:

a) f(x) = x

x−1 b) f(x) = x

x c) f(x) =

√x x

3. For which parameters a₁ and a₂ is the function f : R → R, f(x) = a₁x³ +a₂x² injective and/or surjective? For which parameters isf bijective?

4. Decide, which of the following functions are invertible and which not:

a) f :R−→R, f(x) = cos(x) b) f : [−π,0]−→R, f(x) = cos(x) c) f : [−π,0]−→[−1,1], f(x) = cos(x) Explain your decision!

5. Transform the following terms to make the denominator free of any roots, e. g.

√1 2 =

√2 2 .

Write the numerator in the simplest form you can find.

1 +√ 3 2√

3 + 3√

2 , 1 +√

2−√ 3 1−√

2 +√ 6

√3−√

√ 2 2 +√

3−√ 8 .