Lesson 4.2 Combining Functions; Composite Functions

Key Objectives

  • Understand Functions: Learn how to find sums, differences, products, and quotients of two functions.

  • Average Cost: Be able to form average cost functions based on total cost.

  • Composition of Functions: Learn how to combine two functions to create a new one.

  • Profit Function: Understand how to calculate profit based on costs and revenue.

Example Scenario

For a product:

  • The daily total cost can be written as: C(x) = 360 + 40x + 0.1x² (this means it costs money to produce things based on how many you make).

  • The money you make from selling x units is written as: R(x) = 60x.

Profit Calculation

The profit you make is determined by subtracting costs from revenue:

  • Profit Function: P(x) = R(x) - C(x).

  • If you do the math, it becomes: P(x) = -0.1x² + 20x - 360.

Operations with Functions

  • Adding Functions: (f + g)(x) = f(x) + g(x)

  • Subtracting Functions: (f - g)(x) = f(x) - g(x)

  • Multiplying Functions: (fg)(x) = f(x) * g(x)

  • Dividing Functions: (f/g)(x) = f(x) / g(x)

Example Functions

  • Let’s say you have two functions, f(x) = √x and g(x) = x³. Then:

    • (f + g)(x) = √x + x³

    • (f - g)(x) = √x - x³

    • (fg)(x) = √x * x³ = x⁴.5

Domain of Functions

  • When we talk about the domain, we mean what numbers we can use for our functions. For the sum, difference, and product, you can use any real number as long as the functions are defined there. For division, make sure the denominator isn’t zero!

Revenue, Cost, and Profit Example

  • Demand Function: Price you charge = p(x) = 1000 - 2x

  • Total Revenue: Money from selling x items = R(x) = (1000 - 2x)x = 1000x - 2x²

  • Total Cost: Things you spend to make x items = C(x) = 4x + 1999

  • Profit: P(x) = R(x) - C(x), which turns into: P(x) = -2x² + 996x - 1999.

  • The highest profit happens when x = 249, where you make P(249) = 122,003 dollars.

Unconventional Vehicle Sales Projections

We have different types of vehicle sales predictions:

  • E85 flex fuel: F(x) = -0.0027x² + 0.193x + 0.439

  • Diesel: D(x) = 0.028x + 0.077

  • Hybrid Electric: H(x) = 0.025x + 0.154

  • Mild Hybrid: M(x) = -0.006x² + 0.350x

  • Plug-In Electric: E(x) = -0.001x² + 0.081x - 1.115

  • You can add these functions together for predictions on sales, showing that hybrid/electric vehicles won't sell more than flex fuel/diesel before 2050.

Average Cost Function

  • Average cost per item when making x items: C(x) = C(x) / x

  • For example, if Sunny's Greenhouse has total costs for roses given by: C(x) = 50x + 500, the average cost function would be: T(x) = (50x + 500) / x.

Composition of Functions

  • Function Composition: If you have two functions, then the composition means putting one function inside the other: (f ∘ g)(x) = f(g(x)).

  • The domain depends on the second function, g.

Every Application Matters!

  • Real-life examples include things like converting Celsius to Fahrenheit, which shows how functions can work together!

  • Many practice questions can help you understand function combinations and profitability better!