# 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) = x^{2}, 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(x^{2}) + 2 = 3x^{2} + 2

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

g(x) = x^{2}

g(f(x)) = f(x)^{2} = (3x + 2)^{2} = 9x^{2} + 12x + 4

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

## Handouts/Worksheets

## Function Worksheet

f(x) = 3x + 5 and g(x) = x^{2}+ 2x, h(x) = x + 2, k(x) =

- f(3) =

- f(-1) =

- g(0) =

- g(2) =

- f(4) + g(-2) =

- f(3m) =

- f(n - 2) =

- g(2k) =

- g(a + 2) =

- m(k
^{4}- 5)

- k()1t + 1

- f(g(x)) =

- g(f(x)) =

- k(2) =

- k(0) =

- m(4) =

- m(2k - 2) =

- f(h(g(x)) =

- h(h(x)) =

- k(k(x)) =

- f(m(x)) =

- g(m(x)) - 2m(x) =

- m(g(x) - 4) =

- g(f(x) - h(x)) =

## Function Worksheet: Answer Key

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