1. Solve the following for x. Give exact (not calculator) answers.
(a) e2x+1 = ex,
(b) e2x+1 = 5,
(c) ln(x + 1) = 1,
(d) ln(1 2) + 1 2 ln(x) = 1 2,
(e) ln(1 + 1 x) = −1, (f) loge(x) = 2loge(5) + loge(6)−loge(12).
For Questions 2 and 3, refer to Number Systems Practice Class 3. 2. Let f : R→R and g : R→R be deï¬ned by f(x) = x3 and g(x) = x9 + 1. (a) Give the formula for a function φ : R→R such that g = φâ—¦f. (b) Give the formula for a function ψ : R→R such that g = f â—¦ψ.
3. Let f : R → R be the function deï¬ned by f(x) = 3x2 + 6x, and let g : [−1,0] → R be the restriction of f to [−1,0]. Apply the method used in Question 7(c)(iv) and 9(b)(iii) on Number Systems Practice Class 3 to ï¬nd a formula for g−1(x). Set your answer out carefully and justify any choices you make.
For Question 4, refer to Number Systems Practice Class 4.
4. The logic statement (*) below is the deï¬nition of the fact that the limit of the sequence 1 n equals 0 (from the second year subject MAT2ANA; more on sequences and limits in Number Systems Lecture 6). The symbol R+ denotes the set of positive real numbers. (∀ε ∈R+)(∃N ∈N)(∀n ∈N) n > N =⇒ 1 n −0 < ε. (*)Write the following in symbols (the only English words you may use are “and” and “or”):
(a) the contrapositive of (*);
(b) the converse of (*);
(c) the negation of (*).
For Questions 5 and 6 refer to Practice Classes 3 and 4 of Linear Algebra, Lectures 3 and 4, and Chapter 2.1–2.5 of ‘Notes on Linear Algebra’.
5. (a) For the matrices