Which of the following functions is a one to one function...

Which of the following functions is a one-to-one function? Question 2 options: f: ℤ -> ℤ, where f(n) = |n| - 7 f: ℤ -> ℤ, where f(n) = |2n - 1| f: ℝ -> ℝ, where f(x) = 2x + 7 f: ℝ -> ℝ, where f(x) = 3x^2 + 6

Transcript

00:01 In this problem, we are given three functions, and we want to determine which are onto or which are 1 to 1.

00:07 So obviously, all of these functions are at least onto.

00:10 We have an, if we put in an input, we will get an output.

00:20 But not all of these functions are what to 1 to 1, which means that not all of the outputs are unique per input.

00:27 And functions that are 1 to 1, since they are unique, they can be reversible.

00:31 So we can find their inverse function.

00:34 So let's look at our first function.

00:39 Our first function f is a number that inputs a natural number and outputs two natural numbers.

00:54 Or f will return the coordinate minus the absolute value of x and the absolute value of x.

01:06 So because of the absolute value, we note that that f of, let's say, f of minus x will give us f of x.

01:18 So f of minus x will give us minus the absolute value of minus x and the absolute value of minus x, yielding minus the absolute value of x and the absolute value of x, which means that we have a function that is only onto and not 1 to 1.

01:44 Because let's say we were given this output coordinate, we would not be able to trace what produced this.

01:52 Is it minus x or plus x? we don't know.

01:55 So this here is onto only and not onto 1.

02:08 Next, consider the function g that can receive as input a real number between minus infinity and minus 2 and will output a number between 0 and plus infinity.

02:22 And this function, g of x, is defined as d squared of x squared minus 4.

02:42 So to solve these types of problems, where we have a concrete function, it helps to plot the function.

02:51 It's not necessary, but if you have a graphing calculator, it certainly can help to gain a little bit of intuition.

03:12 So if we were to plot the function, it looks similar to a square root function, but because of the square term, our function is also defined for negative values of x.

03:38 Oh well i suppose, excuse me, it is specified that we only can receive as inputs negative values of x...

Question

Which of the following functions is a one-to-one function? Question 2 options: f: ℤ -> ℤ, where f(n) = |n| - 7 f: ℤ -> ℤ, where f(n) = |2n - 1| f: ℝ -> ℝ, where f(x) = 2x + 7 f: ℝ -> ℝ, where f(x) = 3x^2 + 6

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