Understanding the Modulus Function and its Properties

Standard Functions: Modulus

• Modulus function returns the magnitude (absolute value) of a number.
• If x is positive or zero, |x| = x.
• If x is negative, |x| = -x.
• |5| = 5
• |-5| = -(-5) = 5

Key Modulus Results for Solving Equations and Inequalities

• If |x| = a, then x = a or x = -a.
• If |x| ≤ a, then -a ≤ x ≤ a.
• If |x| ≥ a, then x ≥ a or x ≤ -a.
• These results are crucial for solving equations and inequalities involving modulus.

Common Mistake

• When |x| > some value, consider both positive and negative cases.
• Example: To satisfy magnitude, values should lie between -5 and 5.

Magnitudes of Negative Numbers

• For what values of x is |x| > x
• It is only true for negative numbers, because the absolute value of a negative number is always greater than the number itself.
• The same is also true for the condition |x| ≥ 0, for this to be true x should be real except for zero.

Properties of Modulus

• |a * b| = |a| * |b| (Modulus of a product)
• |a / b| = |a| / |b| (Modulus of a ratio)
• |a + b| ≤ |a| + |b| (Triangle inequality)
• Equality holds in triangle inequality i.e., |a + b| = |a| + |b| when a * b ≥ 0 (a and b have the same sign or are zero).
• Inequality Sign: If a * b < 0, then |a + b| < |a| + |b|.

Solving Equations using Equal Condition in Modulus:

• If |a + b| = |a| + |b|, then a * b >= 0

Standard Functions: Greatest Integer Function (GIF)

• The Greatest Integer Function produces the greatest integer less than or equal to x.
• Denoted as ⌊x⌋ or [x] or GIF(x).
• For example, the GIF of 2.7 is 2, and the GIF of -1.9 is -2.
• If x is already an integer, the GIF(x) = x.

Solving GIF Problems

• Solving involves breaking down the number into different ranges
• E.g., value inside GIF is one and one by two, and the values needed to make it that

Key GIF Properties and Results

• If ⌊x⌋ = k (where k is an integer), then k ≤ x < k+1.
• ⌊x⌋ = k has no solution if k is not an integer.
• In inequalities if ⌊x⌋ > k , then x >= k+1 or ⌊x⌋ < k , then x < k

Adding and Subtracting Intergers Inside GIFs

• Important for definite integrals and periodic functions.
• For integer k, ⌊x + k⌋ = ⌊x⌋ + k.

Dealing with GIF of Negative Numbers -Important for Integration:

• ⌊-x⌋ = -⌊x⌋ if x is an integer; ⌊-x⌋ = -⌊x⌋ - 1 if x is not an integer.
• This handles expression with a floor.

Standard Functions: Fractional Part Function

• Fractional part of x represented by {x} or frac(x).
• Defined as {x} = x - ⌊x⌋.
• The fractional part is always between 0 (inclusive) and 1 (exclusive).

Solving Problems with GIF and Fractional Parts

• First, express {x} in terms of ⌊x⌋.
• Second, because fraction is zero and one in nature
• Use the property 0 ≤ {x} < 1 to find the Integer
• Example - in the form T-1/2.
• You need to show that the value lies between 0 and 1.

Dealing with Fraction of Negative Numers

• If x is an Integer then frac(x) is 0, otherwise
• Then frac(-x) can be written as 1 - frac(x)

Standard Functions: Exponential Functions

• Exponential function graph is very important
• If a power has values around zero and one, the graph goes up
• To solve
  • With equal bases equate the powers
• Inequalities depend on base value.
  • If base>1 do like normal
  • If base 1, then a^x1 > a^x2 if x1 > x2.
• If 0 < a < 1, then a^x1 > a^x2 if x1 < x2 (inequality flips).
• To solve, try to make the base same and solve through inequalities

Standard Functions: Logarithmic Functions

• Logarithmic function graph is very important for base
• If base>1 go up and left
• If base1 keep sign don't worry
• When a