Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x E R | cx + d + 0}. The function f :X → R ax + b is defined by f(x) cx + d 1. Under what condition/s on a, b, c, d will ƒ be one-to-one? 2. In addition to the condition/s above, justify that f is a bijection iff c = 0.

Q: Suppose that the function fis defined, for all real numbers, as follows --x-1 2*-1 (x-1)²-3 f(x)=< 4…

A: we need to find out the values of the function at the given points.

Q: Suppose f and g are non-constant, differentiable, realvalued functions defined on (−∞, ∞).…

A: We plug x=0 and y=0

Q: Compute the convolution L ★ II of the functions L and II, which are defined below: 1 if |æ| <a/2…

A: For the function, Lx and πx, Convolution of Lx and πx can be calculated by L∗πx=∫0x Lt πx-t…

Q: Prove that f(x) = x 3 - 12x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Since, we know  that a polynomial function is always differentiable.⇒f(x)=x3-12x-(i) is always…

Q: If f1(x) and f2(x) are functions from the set of positive integers to the set of positive real…

A:

Q: Suppose that the function f is defined, for all real numbers, as follows. 1 1 if x2 Find f(1), f(2),…

A: f(x) = (−1/4)x − 1 if x≤ −2       = -(x+1)² +3    if −1&lt;x≤2       = −2                 if x&gt;2

Q: f is a function of all real numbers defined by f(x)=x2, what is f-1({x |0<x<1})?

A:

Q: Suppose that the function f is defined, for all real numbers, as follows. X+1 if x2 Find f(-4),…

A: Find the attachment

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: Show that the function g: R - R defined by g(x) = 4x² + 3 is neither one-to-one nor onto

A: To show the function g:R→R defined by gx=4x2+3 is neither one to one nor onto. Solution: First we…

Q: Suppose f  is an odd function and is differentiable everywhere.Prove that for every positive number…

A: Since the function is differentiable everywhere, so the function is continuous too everywhere By…

Q: (a)Verify Rolle's Theorem for f (x)=x - 16x+10 over the interval [0,4].

A:

Q: Prove that the function f(x) = x3-3x2+3x-100 is increasing on R

A: Given function is : f(x) = x3-3x2+3x-100 We have to prove that f(x) is increasing on R

Q: Determine if f(x)=4x√−3x satisfies the Mean Value Theorem on [1,25]. If so, find all numbers c on…

A: Given function: f(x) is as shown on the white board.

Q: Suppose that the function h is defined, for all real numbers, as follows. 1 x-2 if x 2 Find h(-1),…

A:

Q: (b)(is) Verify that the function satisfies the hypotheses of Rolle's Theorem on the given interval.…

A: Explanation of the answer is as follows

Q: If for a real continuous f(x), f(a) f(b) < 0, then in the range of [a,b] for f(x) = 0, there is…

A: It is given that f(x) is a continuous function and f(a)f(b)&lt;0. Then f(a) , f(b) have opposite…

Q: Show that the function g: R -R defined by g(x) = 4x2 +3 is neither one-to-one nor onto

A:

Q: Prove directly from the definition that if X and Y are independent and f and g are measurable…

A: Solution: Given X and Y are independent and f and g are measurable function then we have to prove…

Q: Suppose that the function h is defined, for all real numbers, as follows. 1if x#-1 h (x) = - 2 if x…

A: Topic = Function

Q: Find a function that is defined for all real numbers and has f'(x) that exists for all real numbers…

A:

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: Suppose that the function h is defined, for all real numbers, as follows. 1 *-2 if x2 4 Find h(-1,,…

A: The given function is,h(x)=-14x-2if x&lt;-1(x+1)2-2if -1≤x≤2-14x+2if x&gt;2

Q: ollows. h (x) = 1/4x +1     if x 2 Find (0), h (2), and h (5).

A:

Q: Show that the function f(x)=x4+6x+4 has exactly one zero in the interval [−1, 0].   Which theorem…

A: Intermediate value theorem (IVT) can be used to determine whether a function​ f(x) has any zeros in…

Q: Find a function f and a real number a such that

A:

Q: Suppose that the function f is defined, for all real numbers, as follows. if x<-2 f(x) =< -(x-1)-+2…

A: Choose the function according to the interval

Q: Does there exist a function f such that f(0)  =-1, f(2)  =4, and f'(x) ≤ 2  for all  x?

A:

Q: Suppose that the function f is defined, for all real numbers, as follows. 1 if x 1 Find f(-2),…

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: For the function f(x) = Ax2 + Bx + C, determine the value of c guaranteed by the Mean Value Theorem…

A: f(x) = Ax2 + Bx + C f'(x)=2Ax+B By Mean value theorem:

Q: Suppose that the function f is defined, for all real numbers, as follows. x+1 if x<-1 S) = -(x+1…

A:

Q: If we apply Rolle's Theorem to the function f(x) = 2x- 16x 6 on the interval [2, 6|, how many values…

A:

Q: Find the constant a, or the constants a and b, such that the function is continuous on the entire…

A: fx=3x2,      x≥1ax-4,  x&lt;1 Since fx is a polynomial function for all x&lt;1 &amp; x&gt;1,…

Q: If f (x) = x2 + 2x -1, show thatf (x) satisfies the hypothesis of the Mean Value Theorem on the…

A:

Q: Suppose that the function h is defined, for all real numbers, as follows. --x+5 if x #1 h(x)= 31 -3…

A:

Q: 2. If p and q are positive integers, f is a function defined for positive numbers and attains only…

A:

Q: 1. Define h(x) = +y7x + 2, for all real numbers x > -. (a) Is h a one – to – one function? Why or…

A: Since not a particular question asked ,as per guidelines solution to only first question is given

Q: If c is a domain value for a function f and f(c) = 0, then x = c is called a _____ of thefunction.…

A: c is a domain value and f(c)=0

Q: Find all possible (real number) values of p, q, and r, if any, which result in the following…

A: Given, f(x)=x2-px+6if x&lt;-6px2+x+qif -6≤x≤-4sin(x+4)9(x+4)+qx+rif x&gt;-4

Q: Suppose that the function h is defined, for all real numbers, as follows. h(x)=< 3 if x=-1 -4 if…

A: Topic = Function

Q: Suppose that the function g is defined, for all real numbers, as fo -4 if x < -2 g(x) = (x+1)²…

A:

Q: Determine all the numbers c which satisfy the conclusion of Rolle’s Theorem for the given function…

A: Topic:-  application of derivatives   Function

Q: Suppose that f has zeros at 1 and 3. Show that there are two distinct real numbers x1 and x2 such…

A: answer is in next step

Q: Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval.…

A:

Q: The domain of function x) = In(2– x) is %3D

A:

Q: Show that a function f(x) = ∑ ∞n=2 (1/x-nπ + 1/nπ) is well defined on [0, π]. That is, prove the…

A: Given function,

Q: let f(x)={x^2+2, x >-1 AND -x^2-x, x less than for equal to -1} compute f(0), f(-1)and f(-2) a. 0,…

A:

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.