Functions and Their Graphs

Function Notation and Evaluating

Function notation is a fundamental concept in algebra, representing the output of a function for a given input. The notation f(x) denotes the value of the function when the input is x.

• Definition: If f(x) = 2x^2 - 3x + 1, then to evaluate f(-2), substitute -2 for x and use parentheses for negative values.

• Example:

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

• Fractional Functions: The denominator cannot be zero. Example: has domain , or .

• Radical Functions: The radicand must be non-negative. Example: has domain , or .

• Interval Notation: Brackets [ ] include endpoints; parentheses ( ) exclude them. Infinity always uses parentheses. Union () joins intervals.

Intercepts, Slope, and Reading Graphs

Understanding intercepts and slope is essential for graphing and analyzing functions.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

• Slope:

• Example: For points (1,2) and (3,8):

• Linear Equation:

• Discrete vs. Continuous Graphs: Discrete graphs show separate points; continuous graphs show connected values.

• Vertical Line Test: A graph is a function if no vertical line intersects it more than once.

Function Operations and Domains

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function must exclude values where the denominator is zero.

• Example: If and :

  • ; exclude and

Composition of Functions

Composition involves applying one function to the result of another. The order of composition matters.

• Notation:

• Example: If and :

• Order Matters: in general.

Inverse Functions

An inverse function reverses the effect of the original function. It exists only if the function is one-to-one (passes the horizontal line test).

• Steps to Find Inverse:

  1. Replace with

  2. Switch and

  3. Solve for

  4. Write

• Example: , so

• Check:

Quadratic and Polynomial Functions

Quadratics: Shape and Vertex

Quadratic functions have a characteristic parabolic shape. The vertex is the maximum or minimum point.

• Standard Form:

• Direction: If , parabola opens up; if , opens down.

• Vertex x-coordinate:

• Vertex y-coordinate: Substitute into

• Example: ; , ; vertex

• Axis of Symmetry:

Factoring and Finding Zeros

Factoring polynomials allows us to find their zeros, which correspond to x-intercepts.

• Set and factor.

• Example: ; zeros and

• Factoring by Grouping: ; zeros

Polynomial Graph Clues

The degree and leading coefficient of a polynomial determine its end behavior. Multiplicity affects how the graph interacts with the x-axis.

• Degree: Highest exponent

• Leading Coefficient: Coefficient of highest-power term

• Odd Degree: Ends go opposite directions

• Even Degree: Ends go same direction

• Positive Leading Coefficient: Right end rises

• Negative Leading Coefficient: Right end falls

• Multiplicity: Odd multiplicity crosses x-axis; even multiplicity touches and turns

• Example: : crosses at 3, touches at -1, degree 3

Polynomial Inequalities

Solving polynomial inequalities involves finding critical points and testing intervals.

• Example: ; critical numbers and $1$

• Intervals: , ,

• Test : is true, so solution is

• Strict Inequality: Exclude zeros; for or , include zeros where defined

Polynomial Models and Predictions

Polynomial models are used to fit data and make predictions. The degree of the model should match the pattern in the data.

• Regression: Choose linear, quadratic, or cubic models as appropriate

• Extrapolation: Predictions outside the data range may be unreliable

• Interpretation: Always include units and note limitations

Rational and Radical Expressions

Rational Expressions and Restrictions

Rational expressions are fractions with polynomials in the numerator and denominator. Restrictions are based on denominator zeros.

• Factor: Factor numerator and denominator

• Identify Restrictions: Denominator zeros from the original expression

• Example: ;

• Cancellation: Does not remove original restrictions

Rational Equations

Solving rational equations often involves cross-multiplication and checking for extraneous solutions.

• Example: ;

• Cross-multiply:

• Check that is allowed and satisfies the equation

Rational Function Graph Features

Key features of rational function graphs include asymptotes, intercepts, and holes.

• Vertical Asymptotes: Uncanceled denominator zeros

• Holes: Canceled factors

• Horizontal Asymptotes: Ratio of leading coefficients if degrees are equal

• Example:

  • Vertical asymptote:

  • Horizontal asymptote:

  • x-intercept:

  • y-intercept:

Radicals and Radical Equations

Radical expressions involve roots. The domain is restricted by the radicand, and solving radical equations requires checking for extraneous solutions.

• Domain: For ,

• Solving: ; require

• Square both sides:

• Check candidates: works; does not

Exponential and Logarithmic Basics

Exponential Functions

Exponential functions model growth and decay. The base determines the direction of change.

• General Form:

• Growth:

• Decay:

• Example: models 8% growth per period

• At :

Logarithms Undo Exponentials

Logarithms are the inverse of exponentials. They solve for the exponent given a base and a result.

• Definition: means

• Example: because

• Restrictions: Argument must be positive; base must be positive and not 1

Solving Exponential Equations

Exponential equations can be solved using logarithms.

• Example:

• Take natural logs:

TI-84 Calculator Quick Reference

• Graph a function: [Y=] enter expression → [GRAPH]

• Find zero (x-intercept): [2nd] [TRACE] → 2:zero → left bound, right bound, guess

• Find minimum/maximum: [2nd] [TRACE] → 3:minimum or 4:maximum → left bound, right bound, guess

• Find y at chosen x: [2nd] [TRACE] → 1:value → type x → [ENTER]

• Table of values: [2nd] [WINDOW] (TBLSET) → [2nd] [GRAPH] (TABLE)

• Compare/intersect graphs: Enter expressions in Y1 and Y2 → [GRAPH] → [2nd] [TRACE] → 5:intersect

• Regression: [STAT] → 1:Edit (enter data) → [STAT] → CALC → choose LinReg, QuadReg, or CubicReg

Exam-Day Checklist

• Use parentheses for negative substitutions

• Exclude zero denominators

• Test the sign of each inequality interval

• Check candidate solutions in the original equation

• Distinguish from

• Give interval notation with correct brackets

• Read graph scale correctly

Practice Problems and Answers