Question

Determine whether each of these functions is a bijection from R toR.

Answer 1

Answer:

a) and d) are bijections. b) and c) are not

Step-by-step explanation:

a) Every linear non constant function is a bijection. We can easily find the inverse of f by making a simple calculus.

If y is on the function image, we have y = -3x + 4 for certain x, then

y- 4 = -3x

-(y-4)/3 = x

therefore $f^{-1}(x) = -(x-4)/3$

b) -3 * X² + 4  is not a bijection because quadratic funtions arent bijective. If you evaluate in opposite values you will obtain the same result. For example f(-1) = f(1) = 6

c) (x+1)/(x+2) is not a bijection. It isnt even defined in -2 because the denominator is equal to 0 if X= -2 and we cant divide by 0. A bijective function from R to R must be defined in every element of R. In general, homographic non linear functions are not bijective for the same reason this function is not.

d) $X^5+1$ is bijective. There isnt a simple argument we can use to conclude this. We have no other choice than trying to find the inverse function by making a calculus.

Y = $X^5+1$

Y-1 = $X^5$

$(Y-1)^{1/5} = X$

Not that since 5 is odd, we can calculate $(Y-1)^{1/5}$ independently of which value Y-1 takes. Therefore $f^{-1}(x) = (x-1)^{1/5}$ , and we can conclude that f is bijective.

I hope this helps you!