Ms. Liu Math Final: Unit 1

Types of Numbers

1. Complex Numbers: Numbers in the form a+bia + bia+bi, where aaa is the real part, bbb is the imaginary part, and iii is the imaginary unit (i2=−1i^2 = -1i2=−1).

2. Real Numbers: All numbers on the number line, including rational and irrational numbers.

3. Imaginary Numbers: Numbers that involve iii, e.g., 3i3i3i, where i2=−1i^2 = -1i2=−1.

4. Rational Numbers: Numbers that can be expressed as pq\frac{p}{q}qp​, where ppp and qqq are integers, and q≠0q \neq 0q=0.

5. Irrational Numbers: Numbers that cannot be expressed as pq\frac{p}{q}qp​, e.g., π\piπ, 2\sqrt{2}2​.

6. Integers: Whole numbers, including negative numbers, zero, and positive numbers (…,−2,−1,0,1,2,…\ldots, -2, -1, 0, 1, 2, \ldots…,−2,−1,0,1,2,…).

7. Whole Numbers: Non-negative integers (0,1,2,3,…0, 1, 2, 3, \ldots0,1,2,3,…).

8. Natural Numbers: Positive integers starting from 1 (1,2,3,…1, 2, 3, \ldots1,2,3,…).

Interval and Set Notation

• Interval Notation: Represents subsets of real numbers.

  • (a,b)(a, b)(a,b): Open interval (does not include aaa and bbb).

  • [a,b][a, b][a,b]: Closed interval (includes aaa and bbb).

  • [a,b)[a, b)[a,b) or (a,b](a, b](a,b]: Half-open intervals.

  • (−∞,b)(-\infty, b)(−∞,b): Extends to negative infinity, includes bbb if closed.

• Set Notation: Describes a set of elements using curly braces.

  • Example: {x ∣ x>3}\{x \,|\, x > 3\}{x∣x>3}.

Relations, Functions, and Vertical Line Test

• Relation: A set of ordered pairs (x,yx, yx,y).

• Function: A relation where each xxx-value has exactly one yyy-value.

• Vertical Line Test: A graph represents a function if any vertical line intersects it at most once.

Domain and Range of Different Functions

• Domain: All possible xxx-values (inputs) of a function.

• Range: All possible yyy-values (outputs) of a function.

  • Example: For f(x)=xf(x) = \sqrt{x}f(x)=x​, the domain is [0,∞)[0, \infty)[0,∞), and the range is [0,∞)[0, \infty)[0,∞).

Evaluating Functions

• Substitute the input value into the function.

  • Example: If f(x)=x2+3x−5f(x) = x^2 + 3x - 5f(x)=x2+3x−5, then f(2)=22+3(2)−5=4+6−5=5f(2) = 2^2 + 3(2) - 5 = 4 + 6 - 5 = 5f(2)=22+3(2)−5=4+6−5=5.

Function Composition

• Combining two functions: (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f∘g)(x)=f(g(x)).

  • Example: If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x+1g(x) = x + 1g(x)=x+1, then (f∘g)(x)=f(g(x))=f(x+1)=(x+1)2(f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2(f∘g)(x)=f(g(x))=f(x+1)=(x+1)2.

Application of Function Compositions

• Real-world problems: Compositions model scenarios where one process depends on another.

  • Example: Temperature conversion followed by cost calculation.

Transformation of Functions

1. Vertical Shifts:

   • f(x)+cf(x) + cf(x)+c: Shift upward by ccc.

   • f(x)−cf(x) - cf(x)−c: Shift downward by ccc.

2. Horizontal Shifts:

   • f(x+c)f(x + c)f(x+c): Shift left by ccc.

   • f(x−c)f(x - c)f(x−c): Shift right by ccc.

3. Reflections:

   • −f(x)-f(x)−f(x): Reflect across the xxx-axis.

   • f(−x)f(-x)f(−x): Reflect across the yyy-axis.

4. Stretching and Compressing:

   • af(x)af(x)af(x): Vertical stretch (a>1a > 1a>1) or compression (0<a<10 < a < 10<a<1).

   • f(bx)f(bx)f(bx): Horizontal compression (b>1b > 1b>1) or stretch (0<b<10 < b < 10<b<1).

5. Combinations:

   • Multiple transformations can be applied in sequence.