Suppose f, g are two functions mapping positive real numbers to positive real numbers and f = O(g). Prove why each statement is true or false.
(a) log2 f = O(log2 g)
(b) √f = O(f)
(c) fk + 100fk−1 = O(gk), for k ≥ 1
(a) Let's assume that O[g(n)]
is a composite function, let: g(n) = 5n
O(n) = 4n
And so, ∴ f(n) = O[g(n)] ⇒ f(n) = 20n
However, log2 20n ≠ 4 log2(5n), n ≠ 2 2/3/5
Therefore, the statement log2 f = O(log2 g) is false.
Suppose f, g are two functions mapping positive real numbers to positive real numbers and f...
Suppose that the functions f and g are defined for all real numbers x as follows. f(x) = 4x +1 g(x) = 5x Write the expressions for (f.g)(x) and (f+g)(x) and evaluate (f-g)(-1). (fºg)(x) = 0 (f+8)(x) (6-8)(-1) = 0 o X ?
Need help in proof There are two functions f(x) and g(x) and two real numbers a, b. the period of the function f(x) is T1 and the period of the function g(x) is T2. How do I prove that if T1 and T2 have common multiple, the function y = a*f(x) ± b*g(x) is periodic function and her period is equal to the lowest common multiple of T1 and T2?
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), then f(x)gx)~g(x). TRUE FALSE (c) If the power series Σ an(x + 2)" converges atェ= 5, then it must km0 converge at =-6. TRUE FALSE (d) There exists a power series Σ akz" which converges to f(z)-I on some interval of positive length around FALSE TRUE (e)...
Suppose that the functions f and g are defined for all real numbers x as follows. f(x) = 4x+6 g(x) = x+3 Write the expressions for (g.f)(x) and (g+f)(x) and evaluate (8-8)(3). (9•f)(x) = 1 (+5)(x) = 0 (3-1)(3) = 0 xo?
Let e, f, g, h be real numbers, and suppose that es f. Prove that gse and fshif and only if (e, f) C [g, h]
O GRAPHS AND FUNCTIONS Sum, difference, and product of two functions Suppose that the functions,fand g are defined for all real numbers r as follows. 2 g) 2 and evaluate (g n-. Write the expressions for (g-f)(x) and (g+f(r) and evaluate (g.)(-1). (e11)-D
2. Prove that for any fixed real numbers p and g, the equation 2xr + px+q + log2(x2 + px + q) + x2 + px = 2019 has at most two real number solutions. 2. Prove that for any fixed real numbers p and g, the equation 2xr + px+q + log2(x2 + px + q) + x2 + px = 2019 has at most two real number solutions.
3. In this problem we consider only functions defined on the real numbers R A function f is close to a function g if r e Rs.t. Vy E R, A function f visits a function g when Vr E R, 3y E R s.t. For a given function f and n E N, let us denote by fn the following function: Below are three claims. Which ones are true and which ones are false? If a claim is true,...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3r E R s.t. Vy R, A function f visits a function g when Vz E R,3y E R s.t. < y and lf(y)-g(y)| < We were unable to transcribe this imageBelow are three claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, show...