Pre-Calculus Chapter 1 Test Flashcards

Domain and Range

• Domain refers to the set of all possible input values (x) for a function.
• Range indicates the set of all possible output values (y) resulting from the function.
• The domain typically includes "All Real Numbers" except in certain cases:
  • When the function contains a denominator.
  • When the function involves a square root.

Interval Notation

• Represents sets of numbers using brackets and parentheses, for example, [-3, ∞).

Function Domain Example

• For the function h(x) = √(3 - 2x):
  • The expression inside the square root must be non-negative: 3 - 2x ≥ 0.
  • Solving yields: x ≤ 3/2.

Function Symmetry

• To determine if a function is even, odd, or neither:
  • Replace x with -x in the equation.
  • If all terms remain unchanged, the function is even.
  • If all terms change, the function is odd.
  • If some terms stay the same, the function is neither even nor odd.

Difference Quotient

• This expression, (f(x+h) - f(x))/h, measures the average rate of change of a function.

Slope Concepts

• Slope is defined as the measure of steepness of a line connecting two distinct points.
• Slope Function formula: (∆y)/(∆x) = (y2 - y1)/(x2 - x1).

Line Characteristics

• Parallel lines have the same slope, indicating they run in the same direction.
• Perpendicular lines have slopes that are either negative or reciprocal of one another.

Forms of Linear Equations

• Point-Slope Form: y - y1 = m(x - x1) where m is the slope and (x1, y1) is a point on the line.
• Standard Form: Ax + By = C, where m = -A/B gives the slope.

Secants and Tangents

• A Secant represents a straight line connecting two points on a curve.
• A Tangent represents a straight line that touches a curve at exactly one point.

Average Rate of Change Example

• For the function f(x) = x³ between x1 = 3 and x2 = 4:
  • Calculate f(3) = 27 and f(4) = 64.
  • Then, average rate of change = (64 - 27)/(4 - 3) = 37.

Function Transformations

• Vertical Shift:
  • Upward change described by y = f(x) + c.
  • Downward change described by y = f(x) - c.
• Horizontal Shift:
  • Leftward change described by y = f(x + c).
  • Rightward change described by y = f(x - c).

Reflections

• Reflection across the X-axis represented by y = -f(x).
• Reflection across the Y-axis represented by y = f(-x).