Chapter 4: Additional Topics with Functions

4.1 Transformations of Graphs and Symmetry

This section explores how functions can be transformed and classified based on their symmetry properties. Understanding these concepts is essential for graphing and analyzing functions.

• Even Functions: A function f(x) is even if it is symmetric about the y-axis. For every point (x, y) on the graph, the point (-x, y) is also on the graph. Mathematically: for all x in the domain.

• Odd Functions: A function f(x) is odd if it is symmetric about the origin. For every point (x, y), the point (-x, -y) is also on the graph. Mathematically: for all x in the domain.

• Non-functions and x-axis Symmetry: Graphs that are symmetric about the x-axis are not functions. For every (x, y), the point (x, -y) is also present.

Example: The function is even, while is odd.

4.2 Combinations and Composition of Functions

Functions can be combined using arithmetic operations or composed to create new functions. These operations are foundational for modeling and problem-solving in algebra.

• Combining Functions: Given functions f(x) and g(x):

  • Addition:

  • Subtraction:

  • Multiplication:

  • Division: , where

• Applications: Revenue, cost, profit, and average cost functions are often constructed using combinations of simpler functions.

• Composite Functions: The composition of f and g is written as . The order matters: in general.

Example: If and , then .

4.3 One-to-One and Inverse Functions

This section discusses how to determine if a function is one-to-one and how to find and verify inverse functions. Inverse functions are essential for solving equations and modeling reversible processes.

• One-to-One Functions: A function is one-to-one if each output is produced by exactly one input. It passes the Horizontal Line Test: any horizontal line crosses the graph at most once.

• Inverse Functions: The inverse of a function f, denoted , 'undoes' the action of f. To prove that two functions are inverses, show both and .

• Finding the Inverse: To find the inverse of :

  1. Replace f(x) with y.

  2. Interchange x and y.

  3. Solve for y.

• Symmetry: The graphs of a function and its inverse are symmetric about the line .

Example: For , the inverse is .

4.4 Equations, Inequalities, Radicals, Rational Exponents

This section covers solving equations and inequalities involving radicals, quadratics, and absolute values, using both algebraic and graphical methods.

• Solving Radical Equations: Isolate the radical, then raise both sides to the appropriate power to eliminate it. Always check for extraneous solutions.

• Quadratic Inequalities: Solve by finding the zeros of the quadratic and using sign charts or graphs to determine solution intervals.

• Absolute Value Inequalities: Rewrite as two separate inequalities and solve each.

Example: Solve . Solution: ; expand and solve the resulting quadratic equation.

Chapter 5: Exponential and Logarithmic Functions

5.1 Exponential Functions

Exponential functions model rapid growth or decay and are widely used in science, finance, and statistics.

• General Form: , where , , .

• Exponential Growth: ; the function increases as x increases.

• Exponential Decay: ; the function decreases as x increases.

• Applications: Population growth, radioactive decay, and compound interest.

• Exponential Regression: Fitting an exponential model to data using statistical tools.

• Compound Interest:

  • Annual Compounding:

  • Continuous Compounding:

  Where: = future value, = principal, = interest rate, = time

Example: If , , years, then .

5.2 Logarithmic Functions; Properties of Logarithms

Logarithms are the inverses of exponential functions and are essential for solving equations involving exponents.

• Definition: means , where , , .

• Common Logarithm: Base 10, written as .

• Natural Logarithm: Base , written as .

• Properties of Logarithms:

• Change of Base Formula: , commonly with or .

• Graphing: The graph of passes through (1, 0) and increases slowly for large x.

Example: because .

5.3 Exponential and Logarithmic Equations

This section focuses on solving equations where the variable is in the exponent or inside a logarithm, using algebraic and graphical methods.

• Solving Exponential Equations: If possible, rewrite both sides with the same base and set exponents equal. Otherwise, use logarithms to solve for the variable.

• Solving Logarithmic Equations: Combine logarithms if necessary, then rewrite in exponential form to solve for the variable.

• Change of Base: Useful for evaluating logarithms with uncommon bases.

Example: Solve . Solution: .

5.4 Exponential and Logarithmic Models

Exponential and logarithmic models are used to fit real-world data and interpret trends in various fields such as biology, economics, and engineering.

• Modeling: Use lists and statistical functions on calculators to fit data to exponential or logarithmic models.

• Interpretation: Analyze the meaning of model parameters and predict future values.

Example: Modeling population growth with , where is the initial population and is the growth rate.

5.5 Exponential Functions and Investing

Exponential functions are used to calculate the future or present value of investments with periodic compounding.

• Periodic Compound Interest Formula:

• Variables:

  • = future value

  • = principal (initial investment)

  • = annual interest rate (decimal)

  • = number of compounding periods per year

  • = number of years

• Present Value: Rearranged formula to solve for given .

Example: If , , , , then .

5.7 Logistic and Gompertz Equations

Logistic and Gompertz models describe growth that is limited by environmental factors, commonly used in biology and economics.

• Logistic Growth Model:

  • = carrying capacity

  • = constant determined by initial conditions

  • = growth rate

  • = time

• Gompertz Equation:

  • = upper asymptote (maximum value)

  • = positive constants

• Solving: Both equations can be solved graphically or algebraically for unknowns.

• Interpretation: Understand what each variable represents and how changes affect the model.

Example: In a logistic model with , , , , .