SOLUTION: Please help me with the following Advanced Functions problem as I have tried several times to solve it but have not been able to. Given f(x) = sinx, g(x) = 1/x, and h(x) = |x|

Algebra ->  Functions -> SOLUTION: Please help me with the following Advanced Functions problem as I have tried several times to solve it but have not been able to. Given f(x) = sinx, g(x) = 1/x, and h(x) = |x|       Log On


   

Question 202477: Please help me with the following Advanced Functions problem as I have tried several times to solve it but have not been able to.
Given f(x) = sinx, g(x) = 1/x, and h(x) = |x|
a) State the single equation that represents y = h(g(f(x))).(2 mark)
b) Describe what each composition does in the order that it is applied.(3 marks)
Found 2 solutions by Theo, jsmallt9:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+sin%28x%29
g%28x%29+=+1%2Fx
h%28x%29+=+abs%28x%29
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what is h%28g%28f%28x%29%29%29%3F
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start with the innermost function and work your way out.
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f%28x%29+=+sin%28x%29
g%28f%28x%29%29+=+g%28sin%28x%29%29+=+1%2Fsin%28x%29
h%28g%28f%28x%29%29%29+=+h%281%2Fsin%28x%29%29+=+abs%281%2Fsin%28x%29%29
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f%28x%29+=+sin%28x%29
whatever the value of x is, f(x) is taking that value and replacing it with the sin of that value. f%28x%29+=+sin%28x%29, f%2830%29+=+sin%2830%29, etc.
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g%28x%29+=+1%2Fx
whatever the value of x is, g(x) is taking that value and replacing it with the reciprocal of that value. g%28x%29+=+1%2Fx, g%2830%29+=+1%2F30, g%28f%28x%29%29+=+1%2Ff%28x%29, etc.
if f%28x%29+=+sin%28x%29, then g%28f%28x%29%29+=+1%2Ff%28x%29+=+1%2Fsin%28x%29.
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h%28x%29+=+abs%28x%29
whatever the value of x is, h(x) is taking that value and replacing it with the absolute value of that value. h%28x%29+=+abs%28x%29, h%2830%29+=+abs%2830%29, h%28g%28x%29%29+=+abs%28g%28x%29%29, h%28g%28f%28x%29%29%29+=+abs%28g%28f%28x%29%29%29, etc. if f%28x%29+=+sin%28x%29, and g%28f%28x%29%29+=+g%28sin%28x%29%29+=+1%2Fsin%28x%29, then
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Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
In order to understand functions (and composition of functions in particular) it is very helpful to understand how to interpret the function definition. One should look at the "x" as a placeholder for the input being provided to the function. An example to illustrate: If q(x) = 2x + 3 then function q takes its input, whatever it is and multiplies it by 2 and then adds 3. Consequently:
  • q%28y%29+=+2%28y%29+%2B+3+=+2y+%2B+3
  • q%28x%2B4%29+=+2%28x%2B4%29+%2B+3+=+2x+%2B+8+%2B+3+=+2x+%2B+11
  • q%285x%29+=+2%285x%29+%2B+3+=+10x+%2B+3
  • q%28sin%28x%29%29+=+2%28sin%28x%29%29+%2B+3+=+2sin%28x%29+%2B+3
  • q(q(x)) = q(2x + 3) = 2%282x+%2B+3%29+%2B+3+=+4x+%2B+6+%2B+3+=+4x+%2B+9


Once this is understood, composition of functions becomes fairly simple. In your functions:
f%28x%29+=+sin%28x%29 means function f will return the sin of whatever input you provide.
g%28x%29+=+1%2Fx means function g will return the reciprocal of whatever input you provide it.
h%28x%29+=+abs%28x%29 means function h will return the absolute value of whatever input you provide it.

Now let's look at your composition: h(g(f(x))). Like PEMDAS, we will start with the inner-most expression: f(x). We will replace f(x) with sin(x) to start:
h(g(f(x) = h(g(sin(x))
Working our way out we see g(sin(x)). But the g function always returns the reciprocal of whatever input is provided. So it will return the reciprocal of sin(x). In other "words": g(sin(x)) = 1%2Fsin%28x%29. Subsituting we get:
h(g(sin(x) = h( 1%2Fsin%28x%29 )
And finally, the h function always returns the absolute value of whatever input is provided. So h( 1%2Fsin%28x%29+=+abs%281%2Fsin%28x%29%29. In summary
h(g(f(x))) = h(g(sin(x))) = h( 1%2Fsin%28x%29 ) = abs%281%2Fsin%28x%29%29