Question

please solve the following question and show all the steps. Thank you.

Question 2. We say that a function f is invertible if f--{(ba) : (a, b) function, in which case we call it the inverse function to f. Notice that f} is also a f- (b) = a-> b = f (a) (assuming that f-1 is a function). We define the range of a function f DR to be the set {f(x): r E D], i.e., the set of all values taken by f. a) If f is invertible, what are the domain and range of f in terms of the domain and range of f? b) Which of the following functions are invertible? For those that are invertible, give the inverse. sin x, y) E R2 y-sin(x))

Answer 1

Let f: D rightarrow R be a function then the domain of f is = (x: x elementof D) the range of is (y: y elementof R) if f is the invertible then f^- 1: R rightarrow D the domain of f^- 1 is (y: y elementof R) the range of f^- 1(x: x elementof D) that is the domain of f becomes the range of f^- 1 and the range of f becomes the domain of f^- 1 b) in the given function y = 2x + 1 represents the equation of circle y = 2x + 1 has an inverse since the graph of y = 2x + 1 passes the horizontal line test it we draw a horizontal line mare it up and down then inverse can be found as follows y = 2x + 1 2x + 1 = y x = y - 1/2 therefore f^- 1 (y) = y - 1/2 in the given function c = x^2 + y^2 = 1 represents the equation of a circle with radius 1 unit that is the unit circle the function y = x^2 represents a parabola the parabola