Compose functions and find composite function values. Grade 12 Math 30-1 - exam essential.
Lesson 1.5 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.
A composite function is when you apply one function, and then apply another function to that result.
Think of it like a two-step process:. Step 1: Put x into the first function (inner function)
Step 2: Take that output and put it into the second function (outer function)
We write composite functions as:
f(g(x)) or (f g)(x)
Both mean the same thing: "f of g of x"
How to read it:. Start from the inside → g happens first
Then the outside → f happens second
Example:. f(g(3)) means: "Find g(3) first, then put that answer into f"
Sometimes instead of evaluating at a specific number, we want the general formula.
Example. Given: f(x) = x^2 and g(x) = x + 3
Find: (f g)(x)
Step 1: Write f(g(x))
Step 2: Everywhere you see x in f, replace it with g(x)
f(x) = x^2
f(g(x)) = (g(x))^2
Step 3: Now substitute what g(x) actually is
g(x) = x + 3
f(g(x)) = (x + 3)^2
Step 4: Expand (if needed)
(x + 3)^2 = x^2 + 6x + 9
Answer: (f g)(x) = x^2 + 6x + 9
Given:
f(x) = x + 10 (add 10) g(x) = 2x (double the number)
Find f(g(5)):
Step 1: Work from the inside out - start with g(5)
g(x) = 2x, so:
g(5) = 2(5) = 10
Step 2: Take that result and put it into f
Now we need to find f(10)
f(x) = x + 10, so:
f(10) = 10 + 10 = 20
Answer: f(g(5)) = 20
Now try g(f(5)):
Step 1: Work from the inside out - start with f(5)
f(x) = x + 10, so:
f(5) = 5 + 10 = 15
Step 2: Take that result and put it into g
Now we need to find g(15)
g(x) = 2x, so:
g(15) = 2(15) = 30
Answer: g(f(5)) = 30
⚠️ Notice:. f(g(5)) ≠ g(f(5))
Order matters! You get different answers depending on which function you apply first.
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