Question

Consider the following function f defined over R:

R f(x) = ( log(x) · cos(x) x > 0 0 x ≤ 0 (6)

(a) Determine whether f is one-to-one.
(b) Determine whether f is onto.
(c) Determine whether f is total.
(d) Determine ranges for x and y so that f is total but not one-to-one.
(e) Determine ranges for x and y so that f is one-to-one, onto, and total.

Answer 1

Answer:

a) y=logx*cosx is not One to One/Injective Function b)  y=logx*cosx Not an Onto Function c) y=logx*cosx Not a total Function d)

Step-by-step explanation:

;Assuming this is the function y=logx*cosx  Domain x>0

a) A function is called "one-to-one" if for each value for x1 there is only one value for f(x1), each value for x1 returns a value for fx2 and fx2 $\neq f(x_{2})$

x |  y =logx*cosx

1  |0

2  | 0.3

3  |  0.47

But there's a problem with testing by only constructing a table. There are infinite values. So the Line Test is more efficient than that. If a horizontal line paralell to x-axis crosses the graph in two points.

This function is not One to one.

Check it out below.

b) By the same test, One function is an Onto Function if the horizontal line intercepts only once. An onto function Codomain = Range.

c) Not a total Function

Since the Domain is note the Real Line, there are restrictions over the Domain we cannot affirm this is a Total Function check the Cartesian Plane. Take a good look. All valid entries for x, are over 0 x >0. So This is not a Total one.

d) {y ∈ R |  y>-2} and x ∈ R*

Based on the Horizontal Line Tests

e) x ∈ (0, 1] and  y∈ (-2,0]

Based on the Horizontal Line Tests

Closely looking to the graph

Domain to be total 0<x≤1

Domain One to one, Total and Onto (0, 1]

Range y (-2,0]