for x rational f(x) = for x irrational

Q: 6(x² + 3) ((x – 3)(x + 3))³ 1 -2x f(x) = f (2) = f (x) = (x – 3)(x + 3) ((г — 3)(х + 3))?

A: Since, the numerator is a constant 1, so, the graph has only one horizontal asymptote which is the…

Q: Any rational function is continuous wherever it is defined; that is, it is continuous on its…

A: We know thatPoint No.1: the composition of continuous functions is continuous.Point No.2:…

Q: Given that, f(x) = 2x3 - 3x2 - 12x + 1. F (a) the intervals on which f(x) is increasing

A: Given: f(x)=2x3-3x2-12x+1

Q: Prove that f(x) = (x – 1)2 + (x + 1)? is an even function. A function k(x) is an even function if…

A:

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: Is it possible to determine f (7) if f (x) = 3 for all x < 7 and f is right-continuous at x = 7?…

A:

Q: Show that the following function is continuous or discontinuous at. f(x)=x2+x-6/x2-4

A: The given function is

Q: A function f is said to have a removable discontinuity at (a) if:1. f is either not defined or not…

A: We have to factor the numerator  At x=3 both of the numerator and denominator are zero, so we get…

Q: If f is a rational function, then f is continuous for all real numbers. True O False

A: General form of rational function  p(x)q(x,    where q(x)≠0

Q: 2 For the real-valued functions f (x)=x´- 3 and g (x)= /x-4, find the compositionƒ °g and specify…

A: Given f(x)=x2-3 g(x)=x-4

Q: Consider the function x - 3 for x4 By using the characterization of continuous function in terms of…

A:

Q: snip

A: A function f is decreasing on an open interval I, if for any choice of x1 and x2 in I, with…

Q: (a) Assume that f(x) is a function defined by f(x) = x²-3x+1 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x)…

A: Given.

Q: Define the following real function: f(x) = 2 - x, x is rational and f(x) x2, x is irrational. What…

A:

Q: Suppose that f(x) = 2 for x < 3 and f(x) = -4 for x > 3. (a) What is f (3) if f is…

A: For x<3,f(x) = 2For x>3,f(x) =-4Part (a) - f(3) = ? when f(x) is left continuous.f(3) =…

Q: if x is rational, Consider the function f : R → R defined by f (x) = { otherwise. Determine the set…

A: We know that

Q: Find all values of k that would make the function f(x) continuous at x = such values of k exist,…

A:

Q: Suppose that f: R (0, ∞) and that f'(x) = f(x) ‡ 0. Prove that (f-¹)(x) = 1/x for x > 0.

A:

Q: let be the rational function f be define as R(X) as R(x)=(/sqrt(x)-2)/(x-1) find the values of x…

A: We set R(x)=1 Then simplify

Q: Determine whether the below function f is continuous or discontinuous at a =-3. If f is…

A:

Q: Sketch a function,f(x), with the given conditions: f(x) is not continuous at x=2 lim x→4- f(x) =…

A:

Q: For f(x) = |x-1|, use the definition of differentiability f'(c) = lim determine if f(x) is…

A: Given fx=x-1

Q: A function f(x) is said to have a removable discontinuity at `x-a` if: 1. ` f' is either not defined…

A:

Q: Let f(x)= 1 Use a 8- e proof to show that f is continuous at x = = 2. X-1

A:

Q: Prove that if f(x) is a differentiable function everywhere and f'(x) # 1 for any value of x, then…

A: The objective here is to prove that the mentioned function f(x) has maximum one fixed point. Given:…

Q: Suppose that f: R→R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

A:

Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let f be differentiable on an interval A. If f'(x) not equal to  0 on A,show that f is one-to-one on…

A: Given that, Let f be differentiable on an interval A. If f'(x) not equal to  0 on A. We have to show…

Q: If f (x) = |x − 1| + |x + 3|, then discuss the continuity and differentiability of the function at x…

A: A function f(x) is said to be continuous at a point a if  (1) f(x) is defined at the point a.…

Q: For f(x) = |x - 1|, use the definition of differentiability f'(c) = lim Ax+0 f(c+Ax)-f(c) Ax to…

A: Given: fx=x-1

Q: Is it possible to determine f(7) if f(x) = 3 for all x < 7 and f is rightcontinuous at x = 7?…

A: Solution: Consider the given that f(x) = 3 for all x < 7 f is right continuous at x = 7  The…

Q: Letf(x)= 1/x-1 Use a 8-e proof to show that f is continuous at x = 2.

A:

Q: Show whether f(x) is continuous (3х-2 f(x)={ x-2 x² -4 for x=- for x+ -

A: Given that, f(x) = 3x - 2; for x = - 2 f(x) = (x + 2)/(x^2 - 4); for x not equal to - 2 So when x is…

Q: Show that if f is both even and odd, then f(x) = 0 for every x in the domain of f. %3D

A: Use the defination of a function for to be odd and even.

Q: If f is C2 on an interval prove that f(2 + k) – 2/(2) + {(z =A) = r"(2). - lim h-0 h2 = f"(x). The…

A:

Q: Suppose that f: R R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

A:

Q: Let f(x) = (2x+1)/(x+1)2. (g) Find the intervals of concavity of f, given that…

A: Apply the conditions for a graph  to be concave up or concave down.

Q: Find the open intervals on which f(x) = x3 − 3/2x2 is increasing or decreasing

A: Find the open intervals on which f(x) = x3 − 3/2x2 is increasing or decreasing

Q: Let f (x) = (x-x-2) х-2 1. Show that f (x) is discontinuous at x = 2. 2. Determine whether or not x…

A:

Q: a. f(x) x-2 x-7 b.f(x) = r2+1

A: (a) the given function is f(x)=1x-2 the function is continuous in interval [a,b] if it is continuous…

Q: Determine the intervals on which the following function is continuous. x2 - 3x +2 f(x) = x2 - 1

A:

Q: Give an example of a function f : R → R that is continuous at x if x is an irrational number, but it…

A:

Q: Let f be the following piecewise-defined function. +7 for xs3 f(x) = { 3x +7 for x>3 (a) Is f…

A:

Q: Find the intervals on which f is increasing and the intervals on which it is decreasing. ​f(x)=6−x2

A: To find the function increasing or decreasing If f′(x)>0 on an open interval, then f is…

Q: Let f be the function Зх + 5 for x 5 х Compute the left- and right-hand limits at x = 5 and…

A: Now, Left hand limit at x = 5 is, And right hand limit is,

Q: Determine whether the below function f is continuous or discontinuous at a =-3. If f is…

A:

Q: If f is C2 on an interval prove that f(x + h) – 2f(x) + f(x – h) lim h-0 = f" (x). h2

A: We can prove this with the help of Lagrange Mean value Theorem as follows.

Q: Let f(x) = 2x² + 5x - 7 x-1 = 1 and determine what value for f(1) would make Show that f(x) has a…

A: Solution of given question is written in below steps.

Q: Show that f is a discontinuous function for all x, where f(x) is defined as follows: for x rational…

A: Let us consider a point x= a ,where a is rational, Now for the function to be continuous, at x=a,…

Q: Define g(3) in a way that extends g(x) = (x2 - 9)/(x - 3) to be continuous at x = 3.

A: Here, the given function The function gx is continuous at  x=a, if the following holds: (i) ga is…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.