Function Review

Domain and Range

Understanding the domain and range of a function is fundamental in precalculus. The domain is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) the function can produce.

• Domain: All x-values for which the function produces a real output.

• Range: All y-values that the function can take as x varies over the domain.

• Example: For , the domain is and the range is .

Expanding and FOIL

Expanding expressions, especially binomials, is a key algebraic skill. The FOIL method is used to expand the product of two binomials.

• FOIL: First, Outer, Inner, Last terms are multiplied and then summed.

• Example:

Parabolas

Intercepts and Vertex

A parabola is the graph of a quadratic function, typically written as . Key features include the x-intercepts, y-intercept, and vertex.

• X-intercepts: Points where the graph crosses the x-axis (). Solve .

• Y-intercept: Point where the graph crosses the y-axis (). Compute .

• Vertex: The maximum or minimum point of the parabola. The x-coordinate is .

• Plug the x-coordinate into the function to find the y-coordinate of the vertex.

• Example: For , vertex at , .

Equation from Graph Features

If you know the x-intercepts, y-intercept, and vertex, you can reconstruct the equation of a parabola. Use the general form , where and are the x-intercepts, and solve for using another known point (such as the vertex or y-intercept).

Graphing and Describing Parabolas

• Be able to sketch a parabola using intercepts and vertex.

• Describe the domain (usually all real numbers) and range (depends on whether the parabola opens up or down).

• Example: For , domain is all real numbers, range is .

Rational Functions and Their Graphs

Graphing Rational Functions

A rational function is a ratio of two polynomials, . Their graphs can have asymptotes, holes, and restricted domains.

• Vertical Asymptotes: Occur where and is not also zero.

• Horizontal Asymptotes: Determined by the degrees of and .

• Holes: Occur where a factor cancels from numerator and denominator, indicating a removable discontinuity.

• Example: has a hole at (since cancels), and simplifies to for .

Simplifying Rational Expressions

• Factor numerator and denominator completely.

• Cancel common factors, but first identify values that make the denominator zero (these are excluded from the domain).

• Example: for .

Domain of Rational Functions

• Exclude all x-values that make any denominator zero.

• Example: For , domain is all real numbers except and .

Identifying Asymptotes and Holes

• Vertical Asymptotes: Set denominator equal to zero and solve for x (after simplification, if possible).

• Horizontal Asymptotes: Compare degrees of numerator and denominator:

  • If degree numerator < degree denominator: is the horizontal asymptote.

  • If degrees are equal: .

  • If degree numerator > degree denominator: No horizontal asymptote (may have an oblique/slant asymptote).

• Holes: Occur where a factor cancels from both numerator and denominator.

Polynomial Functions and Roots

Roots, Zeros, and Imaginary Numbers

The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero. For real polynomials, complex roots occur in conjugate pairs.

• Root/Zero: Value such that .

• Real roots: Correspond to x-intercepts on the graph.

• Imaginary roots: If is a root, then is also a root.

• Example: has roots and .

Finding Polynomial Equations from Information

• Given roots, intercepts, or other points, construct the polynomial using the factored form and solve for any unknown coefficients.

Key Theorems and Techniques

• Descartes' Rule of Signs: Predicts the number of positive and negative real roots by counting sign changes in and .

• Rational Root Theorem: Possible rational roots of are .

• Theorem on Bounds: Provides upper and lower bounds for real roots of a polynomial.

• Synthetic Division: A shortcut for dividing a polynomial by a linear factor , useful for testing possible roots.

Calculator and Test Preparation Guidelines

• Graphing calculators are expected for the course and tests.

• One note sheet (one side, full-sized) is allowed during tests; only formulas and written concepts, not worked solutions.

• Practice both with calculators and by hand to prepare for timed tests.

• Tests are 50 minutes; manage your time accordingly.

• Follow all test rules regarding electronics and note sheets.

Study and Practice Recommendations

• Attend class and complete all assignments on time.

• Take notes on homework problems and topics to review.

• Complete review assignments and practice problems, especially those involving multiple concepts.

Additional info: For more detailed examples and practice, refer to homework problems and study guide questions as indicated in your course materials.