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C3 Functions help Watch

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    Hi,
    Can anyone help me with understand the following:

    1) The conditions necessary for the inverse of a function to exist and how to find it (algebraically and graphically)

    2) Understand the functions arcsin, arccos and arctan and their graphs and appropriate restricted domains

    3) Understand what is meant by the terms odd (odd power?), even (even power?) and periodic (trig graphs?)

    Thanks in advance

    ~ Luke
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    (Original post by Luke Collinson)
    Hi,
    Can anyone help me with understand the following:

    1) The conditions necessary for the inverse of a function to exist and how to find it (algebraically and graphically)

    2) Understand the functions arcsin, arccos and arctan and their graphs and appropriate restricted domains

    3) Understand what is meant by the terms odd (odd power?), even (even power?) and periodic (trig graphs?)

    Thanks in advance

    ~ Luke
    1. A function has an inverse if it is both injective and surjective. That is, your function has to pass the 'horizontal line test' (google it)

    3. A function is odd if f(-x) = -f(x) and even if f(-x) = f(x), so symmetric about the y - axis.

    For example f(x) = x^2 is even and f(x) = x^3 is odd; which ties in with your odd/even power thing, but sin is odd and cos is even, those don't have powers.

    A funtion is periodic with period k if f(x+k) = f(x), so trig functions are period (either of period 360/2pi or 180/pi)
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    (Original post by Luke Collinson)
    Hi,
    Can anyone help me with understand the following:

    1) The conditions necessary for the inverse of a function to exist and how to find it (algebraically and graphically)

    2) Understand the functions arcsin, arccos and arctan and their graphs and appropriate restricted domains

    3) Understand what is meant by the terms odd (odd power?), even (even power?) and periodic (trig graphs?)

    Thanks in advance

    ~ Luke
    1) For C3: A function has an inverse only if it is a one-to-one function (or one-one).

    In a one-one function, there is exactly one input to any output. The graph of a one-one function has no turning points (except inflection points), because if it did then there would be (at least) two different values of x that give the same value for y, when y=f(x) is plotted.

    To find the inverse, you can write the formula in the form y=f(x) and rearrange into the form x = g(y), where g is another function, and then rewrite as f^{-1}(x) = ...

    eg, if f(x) = 2x + 3 = y
    \Rightarrow x = \dfrac{y-3}{2}
    \Rightarrow f^{-1}(x) = \dfrac{x-3}{2}

    Graphically, this is the same as reflecting the graph for the function in the line y=x.

    In a many-to-one function, there are at least two inputs for some output. eg f(x) = x^2. You can put in 2 or -2 and get the same value out, so it is many-to-one. The graphs of many-to-one functions will have maximum/minimum points, unless their range is restricted.
    You can't have an inverse of a many-to-one function, because in a function you put in a number and get out another single number, but the inverse of a many-to-one function would be "one-to-many", which doesn't make sense by our definition of "function".

    However, you can have the inverse of a many-to-one function if its domain is restricted such that it is only a one-one function.
    eg f(x) = x^2 for x>0 is now one-one, so you can find the inverse as before.

    2) Look up the shapes of the graph. Note that sin, cos and tan are many-to-one functions, unless their domains are restricted. Only then can you have inverse functions for them. The arcsin etc... graphs are just reflections of the restricted graphs of their inverse function, sin, cos etc..., in the line y=x

    3) Even functions are symmetric about the y axis. f(-x) = f(x)

    Odd functions reflect in the y-axis, but then on one side of the y-axis they reflect with the x-axis too, so that f(-x) = -f(x)

    Periodic functions repeat over a period.
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    (Original post by K-Man_PhysCheM)
    1) For C3: A function has an inverse only if it is a one-to-one function (or one-one).

    In a one-one function, there is exactly one input to any output. The graph of a one-one function has no turning points (except inflection points), because if it did then there would be (at least) two different values of x that give the same value for y, when y=f(x) is plotted.

    To find the inverse, you can write the formula in the form y=f(x) and rearrange into the form x = g(y), where g is another function, and then rewrite as f^{-1}(x) = ...

    eg, if f(x) = 2x + 3 = y
    \Rightarrow x = \dfrac{y-3}{2}
    \Rightarrow f^{-1}(x) = \dfrac{x-3}{2}

    Graphically, this is the same as reflecting the graph for the function in the line y=x.

    In a many-to-one function, there are at least two inputs for some output. eg f(x) = x^2. You can put in 2 or -2 and get the same value out, so it is many-to-one. The graphs of many-to-one functions will have maximum/minimum points, unless their range is restricted.
    You can't have an inverse of a many-to-one function, because in a function you put in a number and get out another single number, but the inverse of a many-to-one function would be "one-to-many", which doesn't make sense by our definition of "function".

    However, you can have the inverse of a many-to-one function if its domain is restricted such that it is only a one-one function.
    eg f(x) = x^2 for x>0 is now one-one, so you can find the inverse as before.

    2) Look up the shapes of the graph. Note that sin, cos and tan are many-to-one functions, unless their domains are restricted. Only then can you have inverse functions for them. The arcsin etc... graphs are just reflections of the restricted graphs of their inverse function, sin, cos etc..., in the line y=x

    3) Even functions are symmetric about the y axis. f(-x) = f(x)

    Odd functions reflect in the y-axis, but then on one side of the y-axis they reflect with the x-axis too, so that f(-x) = -f(x)

    Periodic functions repeat over a period.
    Thank you ever so much. really informative and clearly written. 1 two
 
 
 
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