Explain why the following statement must be True, or provide a counterexample to show that it can be False. If  f  and  g are one-to-one functions defined on all of  R ,  then the function h(x) = f(x)+g(x) is also one-to-one.

Q: Given f (x) =x-4 and g (x)=|x+2|, what is the domain of the quotient function f/g?

A: Domain will be  Whole real numbers except 2 Explanation is given below......

Q: Sketch the graph of a function f that satisfies all the following conditions: (i) f(x) > 0 for all x…

A: Definition used -                i) If f(x)>0 then graph is above x-axis .                ii) If…

Q: Given that f(x) =x-1 and g(x)=x-x, find (f+ g)(6), if it exists.

A:

Q: Suppose that f is a differentiable function such that f(4)=4 and f'(4)=-2. Let g(x)=f(xsquare root…

A: We first find g'(x) and then evaluate g'(4).

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

A:

Q: The function h(x) = (x + 8) can be expressed in the form f(g(x)), where f(x) = x, and g(x) is…

A:

Q: If f and g are both even functions, is f + g necessarily even? If both are odd, is their sum…

A:

Q: Check whether Rolle's theorem conditions are satisfied or not for the given function f(x) = 2x2 – 4x…

A: solution

Q: Draw the graph of the given function. 1. f: R → Z, defined by f(x) = [2x – 3] 2. f : Z → Z, defined…

A:

Q: do you agree with the contention that the functions f(x) = x+2 and g(x)=x2-4 /x-2 are the same in…

A: It is given that, f(x) = (x+2)     and     g(x) = ( x2-4)/(x-2) Then we will prove that f(x) and…

Q: Apply this to the following function: f(x) =2x+3/3x-2 and g(x) = 4x/3x-2(f+g)(x)

A: f(x) =2x+3/3x-2 and g(x) = 4x/3x-2

Q: Let f be the function defined as follows: [3-x, for x <1 f(x)= {ax + bx, 5x-10, for 1<x <2 where a…

A:

Q: Find (fog)(x) and (gof)(x) from the following functions:  • f(x) = 4x + 3 and g(x) = x - 1  • f(x) =…

A: Given:-  • f(x) = 4x + 3 and g(x) = x - 1  • f(x) = 2 - x and g(x) = 3x + 4  • f(x) = 3x + 1 and…

Q: Check the continuity of the functions f given below at 1 and 2. x +1 if x < 1 f(x) = { 2x 1+ x'if x…

A:

Q: Given the following function. - if x2 x > 2 x-2 5, if x = 0 or x = 2 Can f continuous at (-1, ∞)?…

A: Here, for the given function: limx→0-f(x)=limx→01x1+x-1x  (∞-∞…

Q: 3. Find the local extreme points of the following function: f (x) = x3 – f(x) = x³ x² + 2x + 1

A: Differentiating with respect to x, we get Now solving the first derivative by setting it to zero,…

Q: Consider the following function. Find the f(x) = %3D x+3 3x = (x), 1=x %3D

A:

Q: The function h(x) = (x + 8) can be expressed in the form f(g(x)), where f(x) = x, and g(x) is…

A:

Q: Evaluate the indicated function for f(x) = x2 - 2 and g(x) - 1 algebraically. (f - g)(2)

A:

Q: For the function f defined by f(x) = x2 - 4x + 1,

A: Given function: fx=x2-4x+1

Q: 3. (a) Differentiate the function f(x) = x2 from first principles.

A:

Q: The function f(x) = x2 is not one-to-one. Find a suitable restriction on the domain of f so that the…

A: The function is one-one when it maps the distinct elements from the domain to the distinct elements…

Q: For the function f defined by f(x) = 2x*-8x²+10x²-7x+1 for x E R* Use the Ad-hoc Calculations method…

A: Big O notation is a mathematical notation that describes the limiting behavior of a function when…

Q: Assuming f of g = 1/x + 1 and if g(x) is x +1. Find f(x)

A:

Q: . Do you agree with the contention that the function f(x)= x +2 and g(x) = x2 -4/x-2  are the same…

A:

Q: f(x) is a differentaible function. g(x)=[f(x)]^2. If g'(x)/f'(x)=e^x for all x, then what is the…

A: Please see the explanation below

Q: If f(x) is a concave function such that f(x) ≥ 0 for all x then g(x) = ((f(x))1/2 is a concave…

A: We have to evaluate that, '' If f(x) is a concave function such that f(x)≥0 for all x then…

Q: True or false: For the function f(x) =1/(x - 3), f is not defined at 3, so f is discontinuous at…

A: True or false: For the function f(x) =1x-3, f is not defined at 3, so f is discontinuous at 3.

Q: For f(x) = 4x and g(x) = 1/4x, find (f of g)(x) and (g of f)(x). Then determine wether (f of g)(x) =…

A: We have been given functions f(x) = 4x and g(x) = 1/4x. We need to determine if fog(x) = gof(x).

Q: Given that f(0)%3D3, use the graph of f'(x) given below and the Fundamental Theorem of Calculus to…

A: Find the values of f(x).

Q: xe* if x < 0 g(x) = x – 2x2 + x if 0 < x < c , .3 if x 2 c

A:

Q: 3. Determine algebraically whether each of the following functions is one-to-one. Show your work!…

A:

Q: he functions f and g are defined as follows. 3 2 f(x) =x+4x+8 and g(x) = 2. 7

A: Given,        The functions f and g are defined as follows :         fx=x2+4x+8  and  gx=3x2-7

Q: x +1 if x 2 if 1<x<2

A:

Q: show that any function f(x) defined on a symmetrically placed interval can be written as the sum of…

A: Here we have given one function, and we have to prove that the function f(x) can be written as the…

Q: Consider the following functions f (x) = Vr - 1 and g (x) = 2 - x %3D What is the domain of fog (r)

A: Domain of fog(x) is the values of x where inside function g(x )and entire composite function fog(x)…

Q: Do you agree with the contention that the functions f(x) = x + 2 and g(x) = x2 − 4/x − 2 are the…

A: I am not agreed with this fact that f(x) and  g(x) are the same in every respect. The reason behind…

Q: Consider the following functions f (x) =x² – 1 and g (x) = 2 – 2 Evaluate fog(-4)

A: Given data: fx=x2-1 gx=2-x Solution: Calculation of g-4 g(x)=2-x g(-4)=2-(-4) g(-4)=-6

Q: Find all points on the graph of, for which f′(x) = 0 and those for which f′(x) does not exist.

A: Given: f(x) = x2-123  To find: The points on the graph for, which f'(x) = 0 and those for which…

Q: For what value of c is the function below continuous at x=-3? f(x) = { c  when x=-3 1 when x = 1…

A: The function                     f(x)=c                       when x=-31                       when…

Q: Consider the function: f ( x ) = { 3 x + 1 , x  3 Is f ( x ) continuous, not continuous but…

A: Consider the given function individually    f(x)= 3x+1 represents a straight line. The line is a…

Q: The function g (æ) = |x² – 4| is differentiable at x = 4 . - True False

A:

Q: The function h(x) = (x + 6)° can be expressed in the form f(g(x)), where f(x) z, and g(x) is defined…

A: Consider the function hx=x+65 Let fx=x5, then fgx=gx5 Here, the objective is to find the function gx

Q: Explain why the following statement must be True, or provide a counterexample to show that it can be…

A: Let us define f:ℝ→ℝ and g:ℝ→ℝ such thatf(x)=x and g(x)=-xThen both f(x) and g(x) are one to one…

Q: find and simplify f (a +h) -f (a) / h     (h is equal/not equal to 0) for function  F(x) =…

A: Consider the provided question,

Q: Given the function f"(x) = -2x(x – 4)(x – 6), what conclusion can be drawn directly from the sign…

A: The given double derivative function is f"x=-2x3x-4x-6.

Q: Explain why the following statement must be True, or provide a counterexample to show that it can be…

A: Note: A function f from A to B is said to be one-to-one or injective if for all distinct elements in…