Let g : A → B and f : B → C be functions. If f ◦ g : A → C is 1-1, then g is 1-1.(a) Prove this statement.(b) State the converse. Then show that the converse is not true

Q: Let f and g be functions such that f(0) = 3, g(0) = 5, f' (0) = 4, g'(0) = -3. Find h'(0) for the…

A: Given f0=3,    g0=5f'0=4,   g'0=-3hx=gxfx To find h'0

Q: Prove the following statement: If f:X →Y and g:Y → Z are one-to-one functions, then the composition…

A:

Q: Let f and g be functions satisfying that f(1) = 3, g(1) = 4, f'(1) = 1, g'(1) = 5. For the following…

A:

Q: 1 if g is the inverse of function f and f(x) = 1+x? prove that g'(x)=1+{g(x)}"

A:

Q: Let f(x)= {x^2-6+12, if x≤-1/ax+b if x>-1 Find a and b such that the function f(x) is…

A:

Q: Let f1 and f2 be functions from R to R such that f1(x) = x2 and f2(x) = x − x2. What are the…

A: Given : functions f1 , f2 from ℝ to ℝ such that                 f1x=x2    and    f2x=x-x2 To find :…

Q: If f and g are both odd functions, is their product necessarily odd, even or neither? Prove it

A: Given data: *f and g are both odd function. By the definition of the odd function if a function f(x)…

Q: For a function g(x), if there exists x E [a,b] such that g'(x)> 1, then we conclude that g(x) has…

A:

Q: Let f(x) = (x – 1)-2 Show that f(0) = f(2) but there is no number c in (0,2) such that f' (c) = 0.…

A:

Q: There exists a function f such that f(x) > 0, f(x) 0 for all x. True False

A: We are entitled to solve only one question at a time so, I am providing you an answer of the same.…

Q: If f(a) > f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: Definition:

Q: Let g be a function that is defined for all x, x + 2, such that g(3) = 4 and the derivative of g is…

A:

Q: Prove or provide a counterexample to the following statement: if f : A → B and g : B → C are…

A:

Q: 1. Let f be the function defined by f(x)=/x+ 2 . Prove that f is one-to-one .

A: Dear student as per our rule in one session we can only solve the first three subparts of one…

Q: For any function f(r), if g(x) = f(x²) then g(z) is always increasing.

A:

Q: Determine whether there exists a function f : [0, 2] → R or none such that f(0) = -1, f(2)= 4 and…

A: In this question, we need to determine whether there exists any function for the given conditions.

Q: Define functions f and g from R to R by the fol- lowing formulas: For every x E R, 2x° + 2x = 2x and…

A: It is given that, f(x) and g(x) are defined from R to R by the following formulas:

Q: Is the following statement correct? The continuity of the function f does not depend on the choice…

A: Continuity of function

Q: If F is a function from A to B and x is an element of A, then F(x) is ________.

A: It is given that F is the function from A to B.It is known that if A and B are sets and F is a…

Q: Show that f is strictly monotonic on the given interval and therefore has an inverse function on…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Let f(x) and g(x) be functions. Prove, using contradiction method, that if f(g(x)) is one-to-one,…

A: We have , f(x)  and g(x) are two functions.  We know the definition of one-to-one function, suppose…

Q: Suppose f and g are contimuously differentiable functions such that f(x) = d(x) and g(x) f(x) and…

A: Define a new function                                                   r(x) = f2(x) -…

Q: Let f(x)= |x and g(x 3 =x-2. Determine the composite function fo g and its domain. (fo g)(x)=|

A: NOTE: Refresh your page if you can't see any equations. .

Q: Let f(x) and g(x) be defined as follows: f(x) = 2x+ 1, g(x) = /x-3 What is (fog)(x)?

A:

Q: 19. Let ƒ(x) = x/(x-2). Find a function y = g(x) so that (ƒ ∘ g)(x) = x.

A: Given,

Q: and g be functions such that f(0) = 3, g(0) = 3, %3D f'(0) = 4, g'(0) = -2. '(0) for the function…

A:

Q: Let f (x) = x and g (x)=|x| be two real-valued functions, O (x) be a function satisfying the…

A: Introduction :

Q: Let f be the function defined above, where b is a constant. For what values of b, if any, is f…

A: Left limit at 3 should be equal to right limit at 3 for the function to be continuous.

Q: Let f be a function of x. f (x) = (x – 3)4 is an even function. - O True O False

A: Solution: The objective is to determine whether the given function is even or not

Q: State the Mean-value Theorem, and use it to prove that a) if f(x) is differentiable and f'(x) > 0…

A:

Q: Suppose that f and g are functions such that (0) = 4, f'(0) = 5, g(0) = –2, gʻ(0) = 3. %3D æf(x)…

A:

Q: Define functions f, g, and h such that F( x) = 1 /x^2 −1 = ( f x g x h)( x) and the domain of F is…

A:

Q: Suppose thatf and g are functions that are differentiable at x = 1 and that f(1) = 2, f '(1) = -1,…

A: Given :hx = x2+1gx

Q: Let F(x)= f(g(h(x))), where h(0) = 1, h'(0) = 2, g(1) = 3, g'(1) = 4, and f'(3) = 5. Find F'(0).

A:

Q: a) Let f:R → R_be a function defined as f(x) = (x + 2)². Determine whether the function is one to…

A: According to our guideline, we can solve only first question of different question, that is multiple…

Q: Let f and g be functions such that f(0)=4, g(0)=6, f′(0)=6, g′(0)=-4, Find h′(0) for the function…

A: First obtain the derivative of h (x).

Q: Prove that if f(x) and g(x) are both even functions, then h(x) = f(x) + g(x) is an even function.

A: Given that f(x) and g(x) are both even functions.  That means: f(-x)=f(x) and g(-x)=g(x) [by…

Q: For functions f and g, if f and g are both even functions, then f + g is even.

A: For _____ functions f and g, if f and g are both even functions, then f+g is even

Q: Let function k be the function defined by k(x) = i(f(x) ). What is k' (t)

A: Given that,fx=4cos2x-2esin3xfπ=4cos2π-2esin3π=41-2e0=2…

Q: assume f(g(x))=x and f'(x)=1+f2(x) for all x. show that g'(x)=1/(1+x2)

A: Differentiate both sides of f(g(x))=x with respect to x. Use chain rule on the left side.

Q: Suppose f and g are functions with domain R. If both f and g are even but f + g is odd, then prove…

A:

Q: 39. Let f(x) = 2x3 - 4. Find a function y = g(x) so that (f o g)(x) =x+2.

A: We use the definition of composition of functions to find y= g(x).

Q: 5. Prove the following statement: If f:X → Y and g:Y → Z are one-to-one functions, then the…

A: Solution:

Q: Prove the following statements: (a) If f : A → B and g : B → C are one-to-one functions, then go f:…

A: We prove that if f : A → B and g : B → C are one-to-one functions, then go f : A → C is one-to-one.…

Q: Let g be a function that is defined for all x, x # 2, such that g(3)% = 4 and the derivative of g is…

A: To find: Critical points of g where g'(x)=x2-16x-2,x≠2.

Q: Theorem 1. If f : R → R is a C² function, then f is conver if and only if f"(x) > 0 for every x E R.

A: As per the company rule, we are supposed to solve only one problem. Therefore kindly repost the…

Q: If f(a)> f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: If f(a)> f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a relative…

Q: Function f:Z+ Z, defined by f(n) = n – 1 is a one-to-one function. True False

A:

Q: Suppose f and g are functions with domain R. If both f and g are even but ƒ +g is odd, then prove…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Express the function,h(x) =1/(4x + 5) as a composition of two functions f and g so that h(x) = (f ∘…

A: here given as hx=fogx=14x+5

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.