Rivalry2

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Original 2026-01-01

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1 In addition to these types of functions, there are several other important functions used across different mathematical contexts. These are described below:

2 Modulus function: The modulus function outputs the distance of a number from zero on the<a>number line</a>without considering its sign. It is denoted as:

3 \(f(x) = |x|\)

4 This means that whether x is positive or negative, the output is always non-negative. The graph of the modulus function appears in the first and second quadrants because the output \(y = |x|\) is always positive or zero. For example, points (x, y), there is a corresponding point (-x, y), making the graph symmetric with respect to the y-axis.

5 Rational function: A rational function is a fraction made up of two<a>algebraic expressions</a>, typically in the form:

6 \(f(x) = \frac {p(x)}{q(x)}\), where \(q(x) ≠ 0\)

7 A function is undefined at any point where its<a>denominator</a>equals zero. The graph of a rational function often includes asymptotes of horizontal or vertical lines that the graph approaches but never touches.

8 Signum function

9 The signum function (or sign function) indicates the sign of the input number. It doesn’t give a specific value but specifies whether it is positive, negative, or zero.

10 The range of this function is: {-1, 0, 1}

11 If \(x > 0 → f(x) = 1\) If \(x = 0 → f(x) = 0\) If \(x < 0 → f(x) = -1\)

12 This function is commonly used in programming and logic-based applications like conditional branching, sign detection in algorithms, and many more.

13 Even and<a>odd functions</a>: These functions are categorized based on how they behave when the input is replaced by its negative:

14 A function is even if: \(f(-x) = f(x)\)

15 (The graph is symmetric about the y-axis)

16 Examples: \(f(x) = x², f(x) = cos (x)\)

17 A function is odd if: \(f(-x) = -f(x)\)

18 (The graph is symmetric about the origin)

19 Examples: \(f(x) = x³, f(x) = sin (x)\)

20 Periodic function: A function is called periodic if it repeats its values in regular intervals. In simple terms, for some value T, the function satisfies:

21 \(f(x + T) = f(x)\) for all x

22 Trigonometric functions like \(sin (x)\) and \(cos (x)\) are classic examples, repeating every 2π. For instance, \(f(x) = sin (x)\) has a period of 2π and a range of [-1, 1].

23 Inverse function: The<a>inverse function</a>of \(f(x)\), written as \(f⁻¹(x)\), essentially reverses the effect of \(f(x)\). In the inverse:

24 When the domain is \(f(x)\) then the range is \(f⁻¹(x)\)

25 The range of \(f(x)\) becomes the domain of \(f⁻¹(x)\)

26 An inverse function exists only if the original function is bijective (both one-to-one and onto).

27 Example:

28 If \(f(x) = x²\), then the inverse of \(f⁻¹(x) = √x\) (with domain restrictions).

29 For \(f(x) = sin x\), \(domain = ℝ\) and \(range = [-1, 1]\).

30 For \(f⁻¹(x) = sin⁻¹x\), \(domain = [-1, 1]\) and \(range = [-π/2, π/2]\).

31 Greatest integer function: Also known as the<a>floor function</a>, this function maps any<a>real number</a>to the greatest integer less than or equal to it. It is written as:

32 \(f(x) = ⌊x⌋\)

33 Example: \(⌊2.8⌋ = 2, ⌊-1.3⌋ = -2\)

34 The output is always an integer, and the graph appears as a step curve due to its jump at every integer value.

35 Domain: All real numbers (ℝ)

36 Range: Integers (ℤ)

37 Composite function: A composite function is formed by applying one function to the output of another. If f(x) and g(x) are two functions, then the composite function is:

38 \(f∘g(x) = f(g(x))\)

39 In this case, the output of g(x) becomes the input for f(x).

40 Example:

41 Let \(f(x) = 2x + 3\) and \(g(x) = x + 1\)

42 Then, \(f∘g(x) = f(g(x)) = f(x + 1) = 2(x + 1) + 3 = 2x + 5.\)

43 Graphs of different types of functions: Each type of function has a unique graphical representation and must satisfy the vertical line test to be considered a function. The shape of the graph varies based on the type of function.

44 For example:

45 The figure below shows the graphs of various other types of function types. Many of these graphs include horizontal, vertical, or even both types of asymptotes, depending on the nature of the function.