Function Fusion Worksheet
Algebra II • Precalculus Unit 2: Function Analysis
Function Fusion Worksheet
Operations, Linear Combinations, Compositions & Domain Restrictions
Name:
Date:
Period:
Key Rules:
\((f \pm g)(x) = f(x) \pm g(x)\)
\((c \cdot f)(x) = c \cdot f(x)\)
\(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\; g(x) \neq 0\)
\((f \circ g)(x) = f(g(x))\)
1
Part I: Algebraic Operations & Domain
Show work and simplify completely.
Given functions: \(f(x) = 3x - 5\), \(g(x) = x^2 - 4\), \(h(x) = 2x + 1\)
1. Find \((f + g)(x)\)
Answer:
2. Find \((g - h)(x)\)
Answer:
3. Find \((f \cdot h)(x)\)
Answer:
4. Find \(\left(\frac{g}{h}\right)(x)\) & domain restriction
Answer / Domain:
2
Part II: Numerical Evaluations & Linear Combinations
Show substitution and arithmetic steps.
Given functions: \(p(x) = 2x^2 - 3\), \(q(x) = 5 - 2x\), \(r(x) = \sqrt{x + 7}\)
5. Evaluate \((p + q)(3)\)
Value:
6. Evaluate \((p \cdot q)(-2)\)
Value:
7. Evaluate \((r \circ q)(1)\)
Value:
8. Evaluate \(2p(2) - 3q(1) + 4r(q(-2))\)
Value:
3
Part III: Function Decomposition
Identify outer function \(f(x)\) and inner function \(g(x)\) such that \(H(x) = (f \circ g)(x)\).
9. \(H(x) = (4x - 7)^5\)
\(f(x) =\)
\(g(x) =\)
10. \(H(x) = \sqrt{x^2 + 9}\)
\(f(x) =\)
\(g(x) =\)
Function Fusion Worksheet • High School Mathematics Page 1 of 2
Function Fusion Worksheet
• Advanced Compositions & Tabular Analysis
Name:
Score:
4
Part IV: Algebraic Composition & Multi-Term Combinations
Expand and simplify completely in standard form.
Given functions: \(f(x) = 2x - 3\), \(g(x) = x^2 + 1\), \(k(x) = \frac{1}{x + 4}\)
11. Find \((f \circ g)(x)\)
Answer:
12. Find \((g \circ f)(x)\) (Expand fully)
Answer:
13. Find \((k \circ f)(x)\)
Ans / Restr:
14 Find and simplify in standard form:
Featured Challenge
\(2f(x) - 3g(x) + 5f(g(x))\)
Show substitution & distributed steps:
Final Simplified Form: Combine all like terms:
5
Part V: Tabular Evaluation & Multi-Step Calculations
Use the table values below to determine each quantity.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|
| \(f(x)\) | \(4\) | \(1\) | \(-2\) | \(3\) | \(0\) | \(-1\) |
| \(g(x)\) | \(-1\) | \(3\) | \(2\) | \(-2\) | \(1\) | \(0\) |
15. \((f + g)(1)\)
Value:
16. \((f \circ g)(0)\)
Value:
17. \((g \circ f)(2)\)
Value:
18 Evaluate \(2f(1) - 3g(-1) + 5f(g(2))\)
Final Value:
\(2f(1) =\)
\(3g(-1) =\)
\(5f(g(2)) =\)
Function Fusion Worksheet • High School Mathematics Page 2 of 2
Function Fusion Answer Key
Teacher Guide • Solutions Unit 2: Function Analysis
Function Fusion Answer Key
Complete Worked Solutions, Grading Notes & Common Errors
MASTER KEY
Full Credit Criteria Shown
Key Rules:
\((f \pm g)(x) = f(x) \pm g(x)\)
\((c \cdot f)(x) = c \cdot f(x)\)
\(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\; g(x) \neq 0\)
\((f \circ g)(x) = f(g(x))\)
1
Part I: Algebraic Operations & Domain
Standard polynomial form required
Given: \(f(x) = 3x - 5\), \(g(x) = x^2 - 4\), \(h(x) = 2x + 1\)
1. Find \((f + g)(x)\)
\(= (3x - 5) + (x^2 - 4)\)
Answer: \(x^2 + 3x - 9\)
2. Find \((g - h)(x)\) Watch signs
\(= (x^2 - 4) - (2x + 1) = x^2 - 4 - 2x - 1\)
Answer: \(x^2 - 2x - 5\)
3. Find \((f \cdot h)(x)\)
\(= (3x - 5)(2x + 1) = 6x^2 + 3x - 10x - 5\)
Answer: \(6x^2 - 7x - 5\)
4. Find \(\left(\frac{g}{h}\right)(x)\) & domain restriction
\(2x + 1 = 0 \implies x = -\frac{1}{2}\)
Ans / Restr: \(\frac{x^2 - 4}{2x + 1},\; x \neq -\frac{1}{2}\)
2
Part II: Numerical Evaluations & Linear Combinations
Exact integer values
Given: \(p(x) = 2x^2 - 3\), \(q(x) = 5 - 2x\), \(r(x) = \sqrt{x + 7}\)
5. Evaluate \((p + q)(3)\)
\(p(3) = 2(9) - 3 = 15;\; q(3) = 5 - 6 = -1\)
Calculation: \(15 + (-1) = 14\)
6. Evaluate \((p \cdot q)(-2)\)
\(p(-2) = 2(4) - 3 = 5;\; q(-2) = 5 + 4 = 9\)
Calculation: \(5 \cdot 9 = 45\)
7. Evaluate \((r \circ q)(1)\)
\(q(1) = 5 - 2 = 3 \implies r(3) = \sqrt{3 + 7}\)
Calculation: \(\sqrt{10}\)
8. Evaluate \(2p(2) - 3q(1) + 4r(q(-2))\)
\(p(2)=5 \rightarrow 10;\; q(1)=3 \rightarrow 9;\; r(q(-2))=r(9)=4 \rightarrow 16\)
Final Value: \(10 - 9 + 16 = 17\)
3
Part III: Function Decomposition
Accept equivalent non-trivial decompositions
9. \(H(x) = (4x - 7)^5\)
\(f(x) =\) \(x^5\)
\(g(x) =\) \(4x - 7\)
10. \(H(x) = \sqrt{x^2 + 9}\)
\(f(x) =\) \(\sqrt{x}\)
\(g(x) =\) \(x^2 + 9\)
Function Fusion Answer Key • High School Mathematics Page 1 of 2
Function Fusion Answer Key
• Advanced Compositions & Tabular Analysis
TEACHER EDITION • PAGE 2
