Solved Exercises on Combining Functions for Modeling in Algebra 2

Master combining functions: addition, subtraction, multiplication, division, and composition for real-world modeling applications.

Solution: Exercises 1 to 3
1 Function addition
Exercise 1
A company's revenue is modeled by R(x) = 50x - 0.5x² and its cost is modeled by C(x) = 20x + 100, where x is the number of items sold. Find the profit function P(x) and determine the number of items that maximizes profit.
Definition:

Profit function: P(x) = R(x) - C(x), where revenue minus cost equals profit.

Function combination method:
  1. Identify the functions to combine
  2. Apply the operation (addition, subtraction, etc.)
  3. Simplify the resulting expression
  4. Use the combined function to solve the problem
Revenue
R(x) = 50x - 0.5x²
Cost
C(x) = 20x + 100
Profit
P(x) = 30x - 0.5x² - 100
Step 1: Write the profit function

P(x) = R(x) - C(x) = (50x - 0.5x²) - (20x + 100)

Step 2: Simplify the expression

P(x) = 50x - 0.5x² - 20x - 100

P(x) = 30x - 0.5x² - 100

P(x) = -0.5x² + 30x - 100

Step 3: Find the maximum

For a quadratic in form ax² + bx + c, vertex occurs at x = -b/(2a)

Here, a = -0.5, b = 30

x = -30/(2(-0.5)) = -30/(-1) = 30

Step 4: Calculate maximum profit

P(30) = -0.5(30)² + 30(30) - 100 = -0.5(900) + 900 - 100 = -450 + 900 - 100 = 350

P(x) = -0.5x² + 30x - 100
Max profit: 350 at x = 30 items
Final answer:

The profit function is P(x) = -0.5x² + 30x - 100, and maximum profit of $350 occurs when selling 30 items.

Applied rules:

• Function subtraction: Combine like terms

• Quadratic maximum: Vertex at x = -b/(2a) when a < 0

• Profit equation: Revenue minus cost

2 Function composition
Exercise 2
The radius of a circular oil spill grows according to r(t) = 2t + 1, where t is time in hours. The area of a circle is A(r) = πr². Find the composite function A(r(t)) and determine the area after 4 hours.
Definition:

Composite function: (A ∘ r)(t) = A(r(t)), where the output of r(t) becomes the input for A(r).

Inner function
r(t) = 2t + 1
Outer function
A(r) = πr²
Composite
A(r(t)) = π(2t + 1)²
Step 1: Write the composite function

A(r(t)) = A(2t + 1) = π(2t + 1)²

Step 2: Expand the expression

A(r(t)) = π(2t + 1)² = π(4t² + 4t + 1)

A(r(t)) = 4πt² + 4πt + π

Step 3: Calculate area after 4 hours

A(r(4)) = π(2(4) + 1)² = π(8 + 1)² = π(9)² = 81π square units

Step 4: Verify the result

At t = 4: r(4) = 2(4) + 1 = 9

A(9) = π(9)² = 81π ✓

A(r(t)) = π(2t + 1)²
Area after 4 hours: 81π square units
Final answer:

The composite function is A(r(t)) = π(2t + 1)², and the area after 4 hours is 81π square units.

Applied rules:

• Function composition: Substitute inner function into outer function

• Algebraic expansion: (a + b)² = a² + 2ab + b²

• Order of operations: Apply the innermost function first

3 Function division
Exercise 3
The total cost of producing x items is C(x) = 200 + 5x, and the number of items produced is x. Find the average cost function AC(x) = C(x)/x and determine the average cost per item when producing 50 items.
Definition:

Average cost function: AC(x) = C(x)/x, which represents the cost per unit.

Total cost
C(x) = 200 + 5x
Average cost
AC(x) = (200 + 5x)/x
At x = 50
$9 per item
Step 1: Write the average cost function

AC(x) = C(x)/x = (200 + 5x)/x

Step 2: Simplify the expression

AC(x) = 200/x + 5x/x = 200/x + 5

Step 3: Calculate average cost for 50 items

AC(50) = 200/50 + 5 = 4 + 5 = 9

Step 4: Interpret the result

When producing 50 items, the average cost per item is $9.

AC(x) = 200/x + 5
AC(50) = $9 per item
Final answer:

The average cost function is AC(x) = 200/x + 5, and the average cost per item when producing 50 items is $9.

Applied rules:

• Function division: Divide numerator by denominator

• Average calculation: Total divided by quantity

• Domain restriction: x ≠ 0 (can't divide by zero)

Combining Functions Fundamentals
(f + g)(x) = f(x) + g(x)
Function Addition
Addition
(f + g)(x)
f(x) + g(x)
Subtraction
(f - g)(x)
f(x) - g(x)
Multiplication
(f · g)(x)
f(x) · g(x)
Division
(f/g)(x)
f(x)/g(x), g(x) ≠ 0
Composition
(f ∘ g)(x)
f(g(x))
Key definitions:

Function combination: Creating new functions by performing operations on existing functions

Domain of combination: Must consider restrictions from both original functions

Composition: Applying one function to the result of another function

Domain Restrictions: Division requires denominator ≠ 0, composition requires range of inner function to be in domain of outer function
Order Matters: f(g(x)) ≠ g(f(x)) in general (composition is not commutative)
Tip 1: Always consider the domain of the resulting function after combining.
Tip 2: For composition f(g(x)), substitute g(x) wherever you see x in f(x).
Tip 3: Simplify the combined function before using it to solve problems.
Tip 4: Check your work by substituting specific values into the original and combined functions.
Solution: Exercises 4 to 5
4 Function multiplication
Exercise 4
The efficiency of a machine is modeled by E(t) = 0.8 - 0.02t, and the production rate is P(t) = 100 + 5t, where t is time in hours. Find the function representing the effective production rate R(t) = E(t) · P(t) and calculate the effective production rate after 6 hours.
Definition:

Effective production rate: Actual production rate considering efficiency, calculated as efficiency × nominal production rate.

Efficiency
E(t) = 0.8 - 0.02t
Production
P(t) = 100 + 5t
Effective rate
R(t) = (0.8 - 0.02t)(100 + 5t)
Step 1: Write the multiplication

R(t) = E(t) · P(t) = (0.8 - 0.02t)(100 + 5t)

Step 2: Expand using FOIL

R(t) = 0.8(100) + 0.8(5t) + (-0.02t)(100) + (-0.02t)(5t)

R(t) = 80 + 4t - 2t - 0.1t²

R(t) = 80 + 2t - 0.1t²

Step 3: Calculate effective rate after 6 hours

R(6) = 80 + 2(6) - 0.1(6)²

R(6) = 80 + 12 - 0.1(36) = 80 + 12 - 3.6 = 88.4

Step 4: Verify the result

At t = 6: E(6) = 0.8 - 0.02(6) = 0.68

P(6) = 100 + 5(6) = 130

R(6) = 0.68 × 130 = 88.4 ✓

R(t) = 80 + 2t - 0.1t²
R(6) = 88.4 units per hour
Final answer:

The effective production rate function is R(t) = 80 + 2t - 0.1t², and after 6 hours, the effective production rate is 88.4 units per hour.

Applied rules:

• Function multiplication: Multiply each term of first function by each term of second

• FOIL method: First, Outer, Inner, Last terms

• Like terms: Combine similar terms after expansion

5 Multi-step modeling
Exercise 5
A rectangular garden has length L(w) = 2w + 5 and width w, where w is the width in feet. The cost of fencing is C(L,w) = 3(2L + 2w). Find the total cost function in terms of width only and calculate the cost for a garden with width 8 feet.
Definition:

Multi-step modeling: Combining multiple operations: composition followed by multiplication or addition.

Length
L(w) = 2w + 5
Cost function
C(L,w) = 3(2L + 2w)
Substituted
C(w) = 18w + 45
Step 1: Express perimeter in terms of width

Perimeter = 2L + 2w = 2(2w + 5) + 2w = 4w + 10 + 2w = 6w + 10

Step 2: Find the cost function

C(w) = 3(Perimeter) = 3(6w + 10) = 18w + 30

Step 3: Calculate cost for width of 8 feet

C(8) = 18(8) + 30 = 144 + 30 = $174

Step 4: Verify the result

At w = 8: L = 2(8) + 5 = 21

Perimeter = 2(21) + 2(8) = 42 + 16 = 58

Cost = 3(58) = $174 ✓

C(w) = 18w + 30
C(8) = $174
Final answer:

The total cost function is C(w) = 18w + 30, and the cost for a garden with width 8 feet is $174.

Applied rules:

• Substitution: Replace variables with expressions

• Multi-step process: Combine multiple operations in sequence

• Perimeter formula: 2(length + width) for rectangle

Detailed Summary: Combining Functions for Modeling
(f ∘ g)(x) = f(g(x))
Function Composition
Key definitions:

Function addition: (f + g)(x) = f(x) + g(x) - adds outputs of functions

Function subtraction: (f - g)(x) = f(x) - g(x) - subtracts outputs of functions

Function multiplication: (f · g)(x) = f(x) · g(x) - multiplies outputs of functions

Function division: (f/g)(x) = f(x)/g(x) - divides outputs of functions (g(x) ≠ 0)

Function composition: (f ∘ g)(x) = f(g(x)) - applies f to the output of g

Combination Methodology:
  1. Identify functions: Determine which functions need to be combined
  2. Choose operation: Select appropriate combination operation
  3. Apply operation: Perform the mathematical operation
  4. Simplify: Combine like terms and simplify the expression
  5. Apply domain restrictions: Consider any limitations on the input values
  6. Use for modeling: Apply the combined function to solve the problem
Tip 1: For composition f(g(x)), work from the inside out: calculate g(x) first, then apply f.
Tip 2: Always check domain restrictions, especially for division (denominator ≠ 0) and even roots.
Tip 3: Composition is not commutative: f(g(x)) ≠ g(f(x)) in general.
Tip 4: Real-world applications often require multiple combinations to model complex scenarios.
Common errors: Forgetting domain restrictions, incorrect order of operations in composition, arithmetic mistakes in expansion.
Exam preparation: Practice all types of combinations, memorize the order of operations, understand domain considerations.
Essential rules to know:
Operation Notation Formula Restrictions
Addition (f + g)(x) f(x) + g(x) None
Subtraction (f - g)(x) f(x) - g(x) None
Multiplication (f · g)(x) f(x) · g(x) None
Division (f/g)(x) f(x)/g(x) g(x) ≠ 0
Composition (f ∘ g)(x) f(g(x)) Range of g ⊆ Domain of f
Function Combinations: Visual Comparison
Exercise 6: Combined Function Behavior
Compare f(x) = x + 2, g(x) = x - 1, and their combinations: (f+g)(x), (f-g)(x), (f·g)(x), (f/g)(x)

Analysis: The chart shows how different operations combine functions to create new behaviors.

  • Addition: Creates a function that grows faster
  • Subtraction: Creates a constant function
  • Multiplication: Creates a quadratic function
  • Division: Creates a rational function with asymptote

Questions & Answers

Question: I'm confused about function composition. What's the difference between (f ∘ g)(x) and (g ∘ f)(x)? Aren't they the same thing?

Answer: No, they are usually different! Composition is not commutative:

(f ∘ g)(x) = f(g(x)): Apply g first, then apply f to the result

(g ∘ f)(x) = g(f(x)): Apply f first, then apply g to the result

Example: Let f(x) = x² and g(x) = x + 1

  • (f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)² = x² + 2x + 1
  • (g ∘ f)(x) = g(f(x)) = g(x²) = x² + 1

Notice: x² + 2x + 1 ≠ x² + 1, so (f ∘ g)(x) ≠ (g ∘ f)(x).

Think of it like putting on clothes: socks then shoes is different from shoes then socks!

Question: When I combine functions, how do I determine the domain of the new function?

Answer: The domain of a combined function includes only values that work for ALL functions involved:

For (f + g)(x), (f - g)(x), (f · g)(x):

  • Domain = Domain of f ∩ Domain of g (intersection of both domains)
  • All x-values must be valid for both f(x) and g(x)

For (f/g)(x):

  • Domain = {x: x is in both domains AND g(x) ≠ 0}
  • Must exclude values where denominator equals zero

For (f ∘ g)(x):

  • Domain = {x: x is in domain of g AND g(x) is in domain of f}
  • Must ensure g(x) produces values that f can accept

Always check for restrictions like division by zero, square roots of negatives, and logarithms of non-positive numbers.

Question: In exercise 1, why did we get a maximum profit instead of minimum? How do I know which one to expect?

Answer: For quadratic functions, the sign of the leading coefficient determines whether you have a maximum or minimum:

If a > 0 (positive): Parabola opens upward → minimum value at vertex

If a < 0 (negative): Parabola opens downward → maximum value at vertex

In the profit function P(x) = -0.5x² + 30x - 100:

  • The coefficient of x² is -0.5, which is negative
  • This means the parabola opens downward
  • So the vertex represents the maximum value

In business applications, revenue functions often have negative quadratic terms due to market saturation, leading to maximum profit rather than minimum.

Always examine the coefficient of the highest-degree term to determine the direction of the parabola!

Question: How do I know when to use function combination versus just solving each function separately?

Answer: Use function combinations when:

  • Relationships exist: One quantity depends on another (like profit = revenue - cost)
  • Multiple effects: Several factors contribute to an outcome (like total cost = fixed + variable)
  • Sequential processes: One process feeds into another (like radius affecting area)
  • Rate relationships: Efficiency affecting production rate

Solve functions separately when:

  • Each function represents an independent process
  • You need individual values for analysis
  • Comparing separate outcomes

The key is to identify the relationship between the quantities in the problem. If one output becomes an input for another process, you need combination. If they're independent, solve separately.

Real-world modeling almost always involves combinations since systems are interconnected!

Question: What if I need to combine more than two functions? Can I do (f + g + h)(x) or (f ∘ g ∘ h)(x)?

Answer: Yes! You can combine multiple functions:

Addition/Multiplication: These operations are associative, so you can group them any way:

  • (f + g + h)(x) = f(x) + g(x) + h(x)
  • (f · g · h)(x) = f(x) · g(x) · h(x)

Composition: These must be done sequentially, and the order matters:

  • (f ∘ g ∘ h)(x) = f(g(h(x))) - work from inside out
  • First apply h, then g to the result, then f to that result

Example: If f(x) = x², g(x) = x + 1, h(x) = 2x

  • (f ∘ g ∘ h)(x) = f(g(h(x))) = f(g(2x)) = f(2x + 1) = (2x + 1)²

For addition/multiplication, you can combine any number of functions. For composition, you apply them one at a time in sequence.