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Chain Rule

Composite Functions

By substituting one function for the variable of another function we obtain what is known as a composition of functions. For the purpose of this discussion we will deal only with the functions obtained by substituting a function into one of the basic functions (xn, sin x, cos x, ex and ln x). That is, we apply a basic function to the value of another function.

For example, if we substitute the function g(x) = x2 into the variable of the function f(u) = sin(u), we obtain the function f(g(x)) = sin(x2). We can view this as applying the sine function to the function g(x) = x2.

Constructing a composite function is a bit like spraying paint. If we apply one colour to another, the final colour depends on the order of the application of colours before it. If we apply one function to another, we get a new composite function that is dependent on the order of the functions applied before it.

Exercise

Apply any basic function to some given function to obtain new composite functions. Perform any two operations to make new composite functions. Reset to make others.

can to spray onto some variables and simpler composite functions up to two new coloured operations, including sine, cosine, tangent, logarithm and natural logarithm, exponent and raise to a power and hence form new coloured composite functions

Reset

logarithm

sine

natural logarithm

cosine

exponent

tangent

raise to a power

A composite function is sometimes called a function of a function.

In general if we apply a function f to a variable x and then apply another function g to the result we obtain the composite function g(f(x)) or (goff)(x). The notation goff is read as "g circle f" or "g composed with f" or simply "g of f".

Start with x and apply first f then g to obtain g of f of x.  Start with x and apply first g then f to get f of g of x.

The first function taken is the innermost function. The second or subsequent function are the outer functions. Thus we get:

Start with x and apply first f then g then h to obtain h of g of f of x.

Exercise

Given the simple functions, f to p, make up these composite functions.

f(x) = sin x      g(x) = x + 3      h(x) = log x       k(x) = ex      m(x) = x3      p(x) = 1/x

1. fofg(x)
2. koff(x)
3. fofk(x)
4. mofg(x)
5. pofm(x)

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11. p(m(x))
12. p(f(x))
13. h(p(x))
14. k(g(x))
15. m(f(x))

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answers: 1. sin(x+3) 2. e^(sin x) 3. sin (e^x) 4. (x+3)^3 5. X^(−3) 6. 1/(log x) 7. sin((x+3)^3) 8. exp(−x) 9. log((x+3)^3) 10. log(exp((x+3)^3)) 11. x^(−3) 12. 1/(sin(x)) 13. log(1/x)=−log(x) 14. exp(3+x) 15. (sin(x))^3 16. log(exp(3*x)) 17. exp(sin(x+3)) 18. log(sin((x+3)^3)) 19. 3(log(exp(x)))^3 20. (log(exp(x+3)))^3

6. pofh(x)
7. fofmofg(x)
8. pofk(x)
9. hofmofg(x)
10. hofkofmofg(x)

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16. h(m(k(x)))
17. k(f(g(x)))
18. h(m(f(g(x))))
19. m(h(k(x))))
20. m(h(k(g(x))))

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Exercise

Which of the following functions are expressed as composite functions of a basic function with another function?

  1. cos 3y
  2. (log 3) + 5p
  3. esin (5 − 4x)
  4. (log q)(q − 1)
  5. 4s
  6. log (tan u)
  7. ln p / (2p + 3)
  8. sin x cos x
  9. (5 − log z)10
  10. y3 + y + 4         Check all answers (this will open a new window)

Chain Rule Index | Inner and Outer Functions >>

 

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