Can a one-to-one function have a many-to one inverse function?
For example:
f(x) = 2x^ 3/4 + 1

has an inverse:
f-1(x) = [(x-1) / 2]^ 4/3

How come?

    PizzaMaster
    If you have f(x) = x0.5

    Then f-1(x) = x2, but restricted to the domain R+ u {0}, which means the inverse is also 1-1. The negative branch of x2 isn't a reflection of y = x0.5 in the line y = x, so it's not part of the inverse function

      Ohh so you mean that the range of the original is the domain of the inverse?

      And so that's why
      f(x) = 2x^ 3/4 + 1

      does have an inverse:
      f-1(x) = [(x-1) / 2]^ 4/3
      BUT with domain [1, infinity)

        How do we know whether the question wants us to give the dilation from x axis or y axis when given both the original and transformed function.

        For example:
        f(x)= 1/x2
        f1(x)= 5/x2

        I got dilation by a factor of 1/5 from the y axis but the answers say factor of 5 from the x axis. Is my answer still valid?

          PizzaMaster

          Correct that's why you need to restrict the domain of the original function

            snowflake Your answer wouldn't be considered correct, if the coefficient of x2 were 5 ie y = 1 / (5x2) then it'd be correct since that would be a dilation from the y-axis

                Billzene

                Ohh ok but how do we whether to work out it’s dilation from the X or Y axis if it doesn’t say in the question?

                    snowflake by recognition of the form y = a/(1/b(x-h))n + k

                    a = dilation factor from x axis
                    b = dilation factor from y axis
                    h = translation in +ve direction of x axis
                    n = power
                    k = translation in the positive direction of y axis

                    This is similar to turning point form of parabolas

                        5 days later

                        find values of a and b such that a(x+2)+b(x+3)=18x+8 for all values of x
                        if someone could help that would be great

                          chemistry1111 Expand then factorising LHS gives (a+b)x + (2a + 3b) = 18x + 8 --> by comparing coefficients, you obtain a+b = 18 and 2a+3b =8, solve simultaneously for a and b, giving a=46 and b=-28

                            thank you so much that makes sense

                            also this question
                            For the polynomial P(x)=(a+1)x3 + (b-7)x2 +c+5, find values for a,b and c if P(x) has a degree 2, a leading coefficient of 3 and the constant term is -1

                            when factorizing polynomials in methods unit 3/4, what method do most people use vcaa prefer, long division or equating coefficients

                            9 days later

                            Do we need to buy the new Methods 3/4 book for the new study design (2023)?
                            I feel like the textbook questions are the same...

                            4 days later

                            PizzaMaster
                            Howdy, I did QCE Methods, so if I mess this up, I’m really sorry.

                            https://ibb.co/m0Q1xdV

                            For b) may I know what the answers are? I may be able to work backwards 🙂

                            I didn’t do d) because I wasn’t that confident in doing the question.

                            Hope this helps,
                            -jinx_58

                              20 days later

                              Hi jinx_58,

                              Thank you so much for solving!
                              The answer to part b is Domain = [0, 6]; Range = [0, 9/2]
                              The answer to part c is 9/2

                              The graph of y = 5/x -6 is reflected in the x-axis and
                              then in the y-axis. The equation of the final image is:

                              could someone help with this question

                                chemistry1111

                                1. reflected in the x-axis
                                2. reflected in the y-axis

                                **I'm not quite sure if y = (5/x) -6 or if it's y = 5/(x-6) so I'll show the working for both

                                let f[x] = (5/x) -6

                                1. x reflection: -f[x] = f1[x] = -[ (5/x) - 6 ] = -( 5/x ) + 6

                                2. y reflection: f1[-x] = f2[x] = -(5/ [-x] ) +6 = -(5/ x ) +6

                                let g[x] = 5/(x-6)

                                1. x reflection: -g[x] = f1[x] = -[ 5/(x-6) ] = -5/(x-6)

                                1. y reflection: g1[-x] = f2[x] = -5/( [-x] - 6) = -5/( -x - 6)

                                hope that helps!

                                  Hi everyone,

                                  Is (2x-1) the same as (x-1/2)
                                  I mean they both give x= 1/2
                                  But let's say if we had to factorise 12x3 + 20x2 − x − 6
                                  and we get (2x-1)(2x+3)(3x+2)

                                  Can I also write this as:
                                  (x-1/2)(2x+3)(3x+2)?

                                  ALSO can some please help and find the roots of this: 12x3 + 20x2 − x − 6
                                  using rational root theorem

                                  much appreciated!!!