Cogito
Precalculus · Chapter 1 · Lesson 1
Composing Functions
The output of one becomes the input of the next.
11 problems · about 22 minutes · TEKS P.2.A, P.2.B
What this lesson teaches
The student evaluates and interprets composite functions.
- f(g(x)) runs g first, then f.
- The order almost always changes the result.
- When both composites return x, the functions are inverses.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x + 2, g(x) = 5x. f(g(3))?
Answer 17
Why 17.
f(x) = x + 5, g(x) = 3x. f(g(2))?
Answer 11
Why g(2) = 6.
Same functions. g(f(2))?
Answer 21
Why f(2) = 7.
f(x) = x², g(x) = x − 1. f(g(4))?
Answer 9
Why g(4) = 3.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = 2x, g(x) = x/2. f(g(10))?
Answer 10
Why They undo each other.
Are those two functions inverses? 1 for yes, 0 for no.
Answer 1
Why Both composites return x.
Order the steps for evaluating f(g(x)) at a number.
Answer 1. Substitute the number into g 2. Compute the output of g 3. Substitute that output into f
Why The inner function runs first.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsf(x) = x + 4, g(x) = x − 4. f(g(9))?
Answer 9
Why Inverses.
Is f(g(x)) generally the same as g(f(x))? 1 for yes, 0 for no.
Answer 0
Why Order matters.
Inside First: f(x) = x + 1, g(x) = 2x. What is f(g(3))?
Answer 7
Why 7.
The Other Order: Same functions. What is g(f(3))?
Answer 8
Why 8.