Question 1092863: A. Choose a composite trigonometric function of the form h(x) = g(f(x)) where the function g is one of the trigonometric functions and f is any function more complicated than y = x.
Note: An example of an appropriate function h would be h(x) = sin(3x – 4). However, you must pick a function other than this example for this task.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i picked h(x) = tan(x^2 + 2x + 1)
here's how that equation is derived.
you start with f(x) = x^2 + 2x + 1 and g(x) = tan(x).
you create h(x) = g(f(x)).
in g(x), the argument is x.
in g(f(x)), the argument of x is replaced with f(x) which means it is replaced with (x^2 + 2x + 1) because f(x) is equal to x^2 + 2x + 1.
therefore:
g(f(x)) becomes g(x^2 + 2x + 1) and, since g(x) is equal to tan(x), then g(f(x)) becomes tan(x^2 + 2x + 1).
to see how this works, we first start with f(x) = x^2 + 2x + 1.
a graph of this function shows the following:
when x = -5, y = 16
when x = -1, y = 0
when x = 5, y = 36
the graph is shown below:
we then work with g(x) = tan(x).
a graph of this function shows the following:
when x = -5, y = 3.381
when x = -1, y = -1.557
when x = 5, y = -3.381
the graph is shown below:
we then work with h(x) = g(f(x)) which results in h(x) = tan(x^2 + 2x + 1)
a graph of this function shows the following:
when x = -5, y = .301
when x = -1, y = 0
when x = 5, y = 7.75
the graph is shown below:
so what is happening?
we'll take one point to show the relationship.
when x = -5, f(x) = x^2 + 2x + 1 = (-5)^2 + 2*(-5) + 1 = 25 - 10 + 1 = 16
when x = -5, g(x) = tan(x) = tan(-5) = 3.381
when x = -5, h(x) = g(f(x)) = tan(x^2 + 2x + 1) = tan((-5)^2 + 2*(-5) + 1) = tan(25 - 10 + 1) = tan(16) = .301
you can check with your calculator.
when x = -5, calculate f(x) = x^2 + 2x + 1 and you will get 16.
when x = -5, calculate g(x) = tan(x) and you will get 3.380515006 which rounds to 3.381.
this particular solution has no relationship to f(x).
it just shows how the tan(x) function looks on the graph.
when x = -5, calculate h(x) = g(f(x)) = tan(x^2 + 2x + 1) and you will get .300632242 which rounds to .301.
now find tan(16) and you will find that it is also equal to .300632242 which rounds to .301.
what happened is that, when x = -5, the function of g(f(x)) found the result of x = -5 in the function of f(x) and then calculated the tangent of that value.
so, when x = -5, h(x) found the tan of 16 which was the value of f(x) when x = -5.
you can follow the logic with the other points as well.
when x = 5, f(x) = 36)
when x = 5, h(x) = g(f(x)) = g(36) = tan(36) = 7.750470906which rounds to 7.75.
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