SOLUTION: 1 6 guided notes te function operations and composition ... - Free Printable

You've uploaded a worksheet titled “Function Operations and Composition of Functions”, which includes two sample problems:

- Sample Problem 3: Find and evaluate composite functions.
- Sample Problem 4: Decompose a function into a composition of two functions.

Let’s go through each part step-by-step with clear explanations.

---

## ✔ SAMPLE PROBLEM 3: Evaluate Composite Functions

Part (a):

Given:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x - 2 \)

Find: \( (f \circ g)(6) \)

---

🔍 Step-by-Step Solution:

Recall:
> \( (f \circ g)(x) = f(g(x)) \) — meaning you plug \( g(x) \) into \( f \).

So,

1. First, find \( g(6) \):
\( g(6) = 6 - 2 = 4 \)

2. Now plug that result into \( f \):
\( f(g(6)) = f(4) = \sqrt{4} = 2 \)

✔ Answer: \( (f \circ g)(6) = 2 \)

---

Part (b):

Given:
- \( f(x) = 6x - 1 \)
- \( g(x) = \frac{x + 3}{2} \)

Find: \( (g \circ f)(2) \)

---

🔍 Step-by-Step Solution:

Recall:
> \( (g \circ f)(x) = g(f(x)) \) — meaning you plug \( f(x) \) into \( g \).

So,

1. First, find \( f(2) \):
\( f(2) = 6(2) - 1 = 12 - 1 = 11 \)

2. Now plug that into \( g \):
\( g(f(2)) = g(11) = \frac{11 + 3}{2} = \frac{14}{2} = 7 \)

✔ Answer: \( (g \circ f)(2) = 7 \)

---

## ✔ SAMPLE PROBLEM 4: Decompose a Function

This problem asks you to write a given function \( h(x) \) as a composition of two functions \( f \) and \( g \), such that:
> \( h(x) = (f \circ g)(x) = f(g(x)) \)

---

Part (a):

Given:
\( h(x) = (x^3 - 3x)^2 \)

We want to find \( f(x) \) and \( g(x) \) such that \( f(g(x)) = (x^3 - 3x)^2 \)

---

🔍 Strategy:

Look at the “outer” operation — here, it’s squaring something. So let’s let:

- \( g(x) = x^3 - 3x \) → the inside expression
- \( f(x) = x^2 \) → the outside operation

Then:

\( f(g(x)) = f(x^3 - 3x) = (x^3 - 3x)^2 = h(x) \)

✔ Answer:
- \( f(x) = x^2 \)
- \( g(x) = x^3 - 3x \)

---

Part (b):

Given:
\( h(x) = \frac{3}{3x - 5} \)

We want \( h(x) = f(g(x)) \)

---

🔍 Strategy:

Notice the structure: It’s 3 divided by something. The “something” is \( 3x - 5 \). So we can think:

- Let \( g(x) = 3x \) → this gives us the “3x” part
- Then we need to subtract 5 → but wait, we can also let \( g(x) = 3x - 5 \), and then \( f(x) = \frac{3}{x} \)

That’s simpler!

So:

- \( g(x) = 3x - 5 \)
- \( f(x) = \frac{3}{x} \)

Check:

\( f(g(x)) = f(3x - 5) = \frac{3}{3x - 5} = h(x) \) ✔️

✔ Answer:
- \( f(x) = \frac{3}{x} \)
- \( g(x) = 3x - 5 \)

*(Note: You could also choose \( g(x) = 3x \) and \( f(x) = \frac{3}{x - 5} \), but the above is more straightforward and matches the solution shown in your image.)*

---

## 🧠 Summary of Key Concepts:

- Composition of Functions: \( (f \circ g)(x) = f(g(x)) \) — apply \( g \) first, then \( f \).
- Evaluating Composites: Plug the input into the inner function, then plug the result into the outer function.
- Decomposing Functions: Look for an “inner” expression and an “outer” operation. The outer function usually acts on the entire inner expression.

---

## ✔ Final Answers:

Sample Problem 3:

- (a) \( (f \circ g)(6) = 2 \)
- (b) \( (g \circ f)(2) = 7 \)

Sample Problem 4:

- (a) \( f(x) = x^2 \), \( g(x) = x^3 - 3x \)
- (b) \( f(x) = \frac{3}{x} \), \( g(x) = 3x - 5 \)

Let me know if you’d like to practice more problems or see graphical interpretations!