Determine whether each of these functions is a bijection from r tor.

a) f(x) = -3x + 4

b) f(x) = -3x2 +7

c) f(x) = (x+1)/(x+2)

d) f(x) = x5 +1"

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

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) 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 =

Y-1 =

Not that since 5 is odd, we can calculate independently of which value Y-1 takes. Therefore , and we can conclude that f is bijective.

I hope this helps you!

8121 because

130,000 divided by 16 is 8121