Thank you for your help.
However, I meant that the (x-1) and (x+2) are both under the square root sign.
Sorry for the confusion.

No worries, in that case we have a whole new situation. The same principle applies in that x+2 cannot equal zero and everything inside the bracket must be 0 or greater. To be honest, I would convert (x-1)/(x+2) into a partial fraction which would yield 1 - 3/(x+2) which can be obtained through long division. If you graph that you get a hyperbola (I think it's called that). You just gotta find when the graph is greater than or equal to zero. I would strongly suggest graphing it and you'll come to realise that from -2<x<1 the graph is negative. Thus the domain of that function would be the direct opposite, aka your aforementioned answer.

You'll find graphing stuff is always gonna help you and you need to learn and get used to do doing it. Hope that Helps.

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