Which of the following functions from r rightarrow r are one-to-one (injective), onto (surjective), or both (bijective)? prove your answers. a. f(x) = 5x + 4 b. g(x) = 2x^2 - 2 c. h(x)= 1 + 2/x, x notequalto 0.

a) bijective

b) neither surjective or injective

c) injective

Step-by-step explanation:

Since we are looking from real numbers to real numbers, we want the following things

1) We want every real number y to get it. (surjective)

2) We want every y that gets hit to be hit only once. (injective).

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If we have both things then the function is bijective.

a) f(x)=5x+4 this is a line with a positive slope.

That means it is increasing left to right.  Every increasing or even decreasing line is going to hit every real number y.  This is a bijection.

b) g(x)=2x^2-2 is a parabola. Parabola functions always have y's that get hit more than once and not all y's get hit because the parabola is either open up or down starting from the vertex.  This function is neither injective or surjective.

c) h(x)=1+(2/x)  x is not 0.

1+(2/x) is never 1 because 2/x is never 0 for any x. This means the real number y=1 will never be hit and is therefore not surjective. This function is injective because every 1 that is hit is only hit once. If you want use the horizontal line test to see this.

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1