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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

## 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)) =