Function Composition and Fractals

• Fractals are generated through function composition.
• Images of fractals visually represent mathematical models with complexity, self-similarity, and infinite complexity.
• Colors in fractals represent how many times a function needs to be composed with itself to reach a predetermined value.
• Benoit Mandelbrot pioneered the study of fractals in the 1970s.
• Fractals model complex systems and generate realistic computer graphics.
• Fractals help in understanding chaos and dynamical and recursive systems.
• Fractal-like shapes exist in real life, like coastlines, clouds, mountains, and biological structures.
• Function composition simplifies model representation and helps reveal hidden relationships.

Learning Objectives of This Section

• Combine functions using algebraic operations.
• Create new functions by function composition.
• Evaluate composite functions.
• Find the domain of a composite function.
• Decompose a composite function into its component functions.

Cost and Temperature Relationships

• The cost to heat a house depends on the average daily temperature.
• The average daily temperature depends on the particular day of the year.
• C(T) gives the cost to heat a house for a given average daily temperature in T degrees Celsius.
• T(d) gives the average daily temperature on day d of the year.
• Cost = C(T(d)) means the cost depends on the temperature, which depends on the day.

Combining Functions Using Algebraic Operations

• Usual algebraic operations such as addition, subtraction, multiplication, and division can combine functions.
• The operations are performed with the function outputs, defining the result as the output of our new function.

Husband and Wife Income Example

• Let w(y) be the wife's income and h(y) be the husband's income in year y.
• Total income T(y) is defined as T(y) = h(y) + w(y).
• This relationship can be written without reference to a year as T = h + w.
• Difference, product, and ratio functions can be defined for any pair of functions with the same kinds of inputs and outputs.

For Two Functions

• For two functions f(x) and g(x) with real number outputs, new functions can be defined: f+g, f-g, and fg.
• (f+g)(x) = f(x) + g(x), (f-g)(x) = f(x) - g(x), and (fg)(x) = f(x)g(x).

Learning Objectives for Variation Problems

• Solve direct variation problems.
• Solve inverse variation problems.
• Solve problems involving joint variation.

Nicole's Sales Commission Example

• Nicole's earnings depend on the amount of her sales with a 16% commission.
• If she sells a vehicle for $4,600, she will earn $736.
• Her earnings can be found by multiplying her sales by her commission, e = 0.16s

Direct Variation Explained

• A relationship in which one quantity is a constant multiplied by another quantity is called direct variation.
• In direct variation, each variable varies directly with the other.
• The formula y = kxn is used for direct variation, where k is the constant of variation
• k is a nonzero constant greater than zero, called the constant of variation.

How to Solve Direct Variation Problems

• Identify the input (x) and the output (y).
• Determine the constant of variation by dividing y by the specified power of x.
• Use the constant of variation to write an equation for the relationship: y = kxn.

Exponential Growth and Decay Objectives

• Model exponential growth and decay.
• Use Newton's Law of Cooling.
• Use logistic-growth models.
• Choose an appropriate model for data.
• Express an exponential model in base e.

Modeling Exponential Growth and Decay

• In real-world applications, functions model the behavior of real world data.
• For rapid growth, select the exponential growth function y = A0ekt
• A0 equals the value at time zero.
• e is Euler's constant.
• k is a positive constant that determines the percentage of growth.

Doubling Time

• Doubling time can be used in exponential growth functions.
• Doubling time is the time it takes for a quantity to double.
• Wildlife populations, financial investments, biological samples, and natural resources may exhibit growth.

Exponential Decay Model Explained

• The exponential decay model describes a quantity falling rapidly toward zero.
• The form of the equation is y = A0ekt, where A0 is the starting value, e is Euler's constant, and k is a negative constant.

Half-Life

• The exponential decay model calculates half-life.
• Half life is the time it takes a substance to exponentially decay to half of its original quantity and includes radioactive isotopes.

Characteristics of the Exponential Function

• An exponential function with the form y = A0ekt exhibits:
• One-to-one function.
• Horizontal asymptote: y=0
• Domain: (-∞, ∞)
• Range: (0, ∞)
• No x-intercept
• y-intercept: (0, A0)
• Increases when k > 0, and decreases when k < 0.

Learning Objectives for Exponential Models

• Build an exponential model from data.
• Build a logarithmic model from data.
• Build a logistic model from data.

Building an Exponential Model from Data

• Regression analysis is a modeling technique that finds a curve that models data collected from real-world observations.
• The model is used to make predictions about future events.

Exponential Regression

Use exponential functions to model exponential growth and decay

• y=abx is the from to use for exponential regression
• Assume a>0.
• b, b must be greater than zero and not equal to one.
• The initial value of the model is y=a.
• If b>1, then the equation shows exponential growth
• If 0<b<1 the function models exponential decay

Correlation Coefficient

• As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable r, r, or r2

How To Perform Exponential Regression Using a Graphing Utility

• Use the STAT then EDIT menu to enter given data.
• Clear any existing data from the lists.
• List the input values in the L1 column.
• List the output values in the L2 column.
• Graph and observe a scatter plot of the data using the STATPLOT feature.
• Use ZOOM [9] to adjust axes to fit the data.
• Verify the data follow an exponential pattern.
• Find the equation that models the data.
• Select "ExpReg” from the STAT then CALC menu.
• Use the values returned for a and b to record the model,
• Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

Residuals Explained

• A residual is the difference between the observed value and a model's predicted value.
• Always subtract actual predicted, in that order.
• Residuals above the line are positive, and residuals below the line are negative.