Polynomial Functions

Definition and Structure

A polynomial function in the variable x is a function of the form:

• The coefficients are real numbers, and n is a non-negative integer (the degree of the polynomial).

• A power function is a special case: .

To determine if an expression is a polynomial, check for integer exponents and real coefficients.

Properties of Even Power Functions

• The graph is symmetric with respect to the y-axis.

• The only intercept is at (0, 0).

• If , the function decreases on and increases on .

• If , the function increases on and decreases on .

• Domain: ; Range: for , for .

Properties of Odd Power Functions

• The graph is symmetric with respect to the origin.

• The only intercept is at (0, 0).

• If , the function increases everywhere; if , it decreases everywhere.

• Domain and range: .

Transformations of Polynomial Graphs

• Shifting, reflecting, and stretching/compressing graphs can be used to obtain new functions from basic forms.

• Example: is a shift of right by 1 and down by 2.

Zeros and Multiplicity

• A zero of satisfies .

• If can be factored as , then is a zero of multiplicity .

• If is even, the graph touches the axis at ; if is odd, the graph crosses the axis at .

End Behavior

• Determined by the leading term .

• For even degree, both ends go up (if ) or down (if ).

• For odd degree, ends go in opposite directions.

Rational Functions

Definition and Domain

A rational function is a function of the form:

• where and are polynomials and .

• The domain is all real numbers except where .

Vertical Asymptotes

• A vertical asymptote occurs at if and .

• As approaches , increases or decreases without bound.

Horizontal and Oblique (Slant) Asymptotes

• Horizontal asymptotes describe the behavior as .

• Let = degree of numerator, = degree of denominator:

  • If , is the horizontal asymptote.

  • If , is the horizontal asymptote.

  • If , there is no horizontal asymptote; instead, there may be an oblique asymptote found by polynomial division.

Graphing Rational Functions: Step-by-Step Strategy

1. Find the y-intercept (set ).

2. Find x-intercepts (set ).

3. Find vertical asymptotes (set denominator to zero).

4. Find horizontal or oblique asymptotes.

5. Plot points between and beyond intercepts and asymptotes.

6. Sketch the graph using all information.

Function Composition and Inverse Functions

Composition of Functions

• The composition is defined as .

• The domain of is all such that $x$ is in the domain of and is in the domain of .

One-to-One Functions

• A function is one-to-one if implies .

• Graphically, every horizontal line intersects the graph at most once (Horizontal Line Test).

Inverse Functions

• If is one-to-one, its inverse satisfies .

• To find the inverse:

  1. Write .

  2. Interchange and to get .

  3. Solve for to get .

  4. Check that and .

Exponential Functions

Definition and Properties

• An exponential function is , where and .

• If :

  • Domain: ; Range:

  • Function is increasing

  • Horizontal asymptote at as

• If :

  • Domain: ; Range:

  • Function is decreasing

  • Horizontal asymptote at as

The Natural Exponential Function

• The number is defined as .

• The function is called the natural exponential function.

Logarithmic Functions

Definition and Properties

• The logarithmic function with base , is .

• It is the inverse of the exponential function .

• Key relationships:

  • if and only if

• If :

  • Domain: ; Range:

  • Function is increasing

  • Vertical asymptote at

• If :

  • Domain: ; Range:

  • Function is decreasing

  • Vertical asymptote at

• The natural logarithm is .

Logarithm Properties and Rules

• Change of base:

Solving Exponential and Logarithmic Equations

Exponential Equations

• To solve , rewrite both sides with the same base if possible, or use logarithms.

• Example: ; since , set .

Logarithmic Equations

• To solve , rewrite in exponential form: .

• Check for extraneous solutions (domain restrictions).

Applications: Financial Models and Growth/Decay

Compound Interest

• For n compounding periods per year:

• For continuous compounding:

• P = principal, r = annual rate (decimal), t = years, n = periods/year

Exponential Growth and Decay

• General model:

• If , models growth; if , models decay.

• = initial amount, = growth/decay constant, = time

Radioactive Decay and Half-Life

• Half-life: time for half a sample to decay.

• Decay model: , with .

Newton's Law of Cooling

• Temperature model:

• C = surrounding temperature, = initial object temperature,

Example: If a bottle of juice at F is placed in a F fridge, and after 10 minutes it's F, use Newton's Law to model and predict future temperatures.

Additional info: The notes cover all key aspects of polynomial and rational functions, function composition and inverses, exponential and logarithmic functions, their properties, and applications in finance and science, as outlined in Precalculus chapters 4 and 5.