Section 7.1: Combining Functions

Overview of Function Operations

Combining functions is a foundational concept in precalculus, involving operations such as addition, subtraction, multiplication, division, and composition of functions. Understanding how to combine functions allows for the construction of more complex mathematical models.

• Function Addition/Subtraction: ,

• Function Multiplication/Division: , (where )

• Function Composition:

Example: If and , then and .

Section 7.2: Power Functions

Definition and Properties

Power functions are functions of the form , where and are constants. They serve as a basis for understanding polynomial and rational functions.

• Even Power Functions: Functions like are symmetric about the y-axis.

• Odd Power Functions: Functions like are symmetric about the origin.

• End Behavior: The degree and sign of determine the end behavior of the graph.

Example: is an even power function; its graph opens upwards and is symmetric about the y-axis.

Section 8.1-2: Polynomial Functions

Characteristics and Graphs

Polynomial functions are sums of power functions with non-negative integer exponents. Their graphs are smooth and continuous, and their degree determines the number of turning points and end behavior.

• General Form:

• Degree: The highest exponent in the polynomial.

• Zeros: The values of where ; these are the x-intercepts.

• End Behavior: Determined by the leading term .

Example: is a cubic polynomial with degree 3.

Section 8.3-4: Rational Functions

Definition and Asymptotic Behavior

Rational functions are quotients of two polynomials, , where . Their graphs can have vertical and horizontal asymptotes, and sometimes oblique asymptotes.

• Vertical Asymptotes: Occur where and .

• Horizontal Asymptotes: Determined by the degrees of and .

• Oblique Asymptotes: Occur when the degree of is one more than the degree of .

Example: has a vertical asymptote at .

Section 1.1: Periodic Functions

Definition and Properties

Periodic functions repeat their values in regular intervals. The most common examples are trigonometric functions such as sine and cosine.

• Period: The smallest positive value such that for all .

• Amplitude: The maximum absolute value of the function from its mean position.

• Frequency: The number of cycles per unit interval.

Example: has period and amplitude $1$.

Section 1.2-1.5: Trigonometric Functions

Basic Properties and Applications

Trigonometric functions relate angles to ratios of sides in right triangles and are fundamental in modeling periodic phenomena.

• Definitions: , ,

• Unit Circle: The coordinates represent the terminal point of angle .

• Special Angles: and their radian equivalents.

Example: ,

Section 1.4: Inverse Trigonometric Functions

Solving for Angles

Inverse trigonometric functions allow us to determine angles from known ratios. They are denoted as , , and .

• Principal Values: Each inverse function has a restricted range to ensure it is a function.

• Multiple Solutions: On the unit circle, many angles can have the same sine, cosine, or tangent value.

• Applications: Used to solve triangles and model periodic phenomena.

Example: or

Section 2.1-2.2: Transformations on Sine and Cosine

Graphical Transformations

Transformations alter the amplitude, period, phase shift, and vertical shift of sine and cosine functions. The general form is .

• Amplitude:

• Period:

• Phase Shift:

• Vertical Shift:

Example: has amplitude $3\pi\frac{\pi}{2}.

Additional info: Some sections referenced external web resources for further examples and practice, but the main concepts are covered above for self-contained study.