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Composite Functions and Function Values

Lesson Plans > Mathematics > Algebra > Functions
 

Composite Functions and Function Values

When I explain functions to students, among all the information I give them about domain and ranges, I tell them: A function is simply a rule. You put a number in, the function performs some actions on that number according to the rule, and spits out a new number.

So if the function is f(x) = 3x + 2, the rule is "whatever number you put into the function, multiply it by three and add two."

f(1) means "the result when you put 1 into the function." Since the rule is "multiply by 3 and add two," the result can be shown like this:

f(x) = 3x + 2
f(1) = 3(1) + 2 = 3 + 2 = 5

I encourage my students to always write the rule again before performing operations on it. It helps them to visualize that whatever is in the parentheses replaces x in the rule.

We also look at things like f(k + 1). How do you evaluate that?

f(x) = 3x + 2
f(k + 1) = 3(k + 1) + 2 = 3k + 3 + 2 = 3k + 5

Doing several of these types of problems prepares students for composing functions:

If f(x) = 3x + 2 and g(x) = x2, what is f(g(x))?

Again, I encourage the students to write the rule out, and then write it with the replacement:

f(x) = 3x + 2
f(g(x)) = 3g(x) + 2
f(g(x)) = 3(x2) + 2 = 3x2 + 2

Then they do the reverse: g(f(x))

g(x) = x2
g(f(x)) = f(x)2 = (3x + 2)2 = 9x2 + 12x + 4

The following is a worksheet that gives each of the types of problems shown above.

Lesson by Mr. Twitchell

Handouts/Worksheets

Function Worksheet

f(x) = 3x + 5 and g(x) = x2 + 2x, h(x) = x + 2, k(x) =
1
x
, m(x) =
x + 5
. Find the following.

  1. f(3) = 
     
  2. f(-1) = 
     
  3. g(0) = 
     
  4. g(2) = 
     
  5. f(4) + g(-2) =
     
  6. f(3m) =
     
  7. f(n - 2) = 
     
  8. g(2k) = 
     
  9. g(a + 2) = 
     
  10. m(k4 - 5)
     
  11. k(
    1
    t + 1
    )
     
  12. f(g(x)) = 
     
  13. g(f(x)) =
     
  14. k(2) = 
     
  15. k(0) =
     
  16. m(4) = 
     
  17. m(2k - 2) = 
     
  18. f(h(g(x)) = 
     
  19. h(h(x)) = 
     
  20. k(k(x)) = 
     
  21. f(m(x)) = 
     
  22. g(m(x)) - 2m(x) =
     
  23. m(g(x) - 4) =
     
  24. g(f(x) - h(x)) =

Function Worksheet: Answer Key

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