Real Functions: Conditions & Properties

Real functions are a fundamental concept in mathematics that describes the relationship between inputs and outputs, where both are real numbers. In other words, a real function is a rule that assigns to each real number in the domain (input) a unique real number in the range (output).

• Real functions can be expressed using a variety of mathematical notation, such as equations, formulas, graphs, and tables. 
• They can be classified into different types based on their properties, including continuous, differentiable, monotonic, periodic, and more.

Key Terms: Real Numbers, Relations, Functions, Domain, Range, Period, Symmetry

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Real Functions

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A real function is a function whose domain and range are both subsets of the real numbers. In other words, a real function is a function that takes real numbers as inputs and produces real numbers as outputs.

Real functions are often used in mathematical models to describe real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of an electrical circuit.

Examples of real functions include polynomial functions, trigonometric functions, exponential functions, logarithmic functions, and many others. Here are some examples:

f(x) = x² - 2x + 1

This is a polynomial function of degree 2. Its domain and range are both subsets of real numbers.

g(x) = sin(x)

This is a trigonometric function. It takes real numbers as inputs and produces real numbers between -1 and 1 as outputs.

h(x) = e^x

This is an exponential function. It takes real numbers as inputs and produces positive real numbers as outputs.

k(x) = ln(x)

This is a logarithmic function. It takes positive real numbers as inputs and produces real numbers as outputs.

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Conditions to be a Real Function

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For a function to be considered a real function, it must satisfy two conditions:

• The domain of the function must be a subset of the real numbers.

The domain of a function is the set of all possible input values for the function.

• For a function to be a real function, its domain must be a subset of the real numbers. 
• This means that the input values of the function must be real numbers, which include all rational and irrational numbers.

For example, the function f(x) = √x is a real function because its domain is the set of non-negative real numbers (i.e., {x | x ≥ 0}). 

However, the function g(x) = 1/x is not a real function because its domain is the set of all real numbers except zero (i.e., {x | x ≠ 0}).

• The range of the function must be a subset of the real numbers.

The range of a function is the set of all possible output values for the function. 

• For a function to be a real function, its range must be a subset of the real numbers. 
• This means that the output values of the function must also be real numbers.

For example, the function f(x) = x² is a real function because its range is the set of non-negative real numbers (i.e., {y | y ≥ 0}). However, the function g(x) = √-x is not a real function because its range is the set of all imaginary numbers (i.e., {y | y = ix, i is the imaginary unit}).

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Relations and Functions

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A relation is a set of ordered pairs that establishes a connection between elements of two sets. A function is a special type of relation in which each element in the domain is paired with exactly one element in the range.

Examples –

1. Relation: {(1,2), (3,4), (2,3), (4,5)}

This is a relation between the set {1,2,3,4} and {2,3,4,5}. It establishes a connection between elements of the two sets.

2. Function: f(x) = x²

This is a function that maps each real number x to its square, resulting in a range of non-negative real numbers. For example, f(2) = 4, f(-3) = 9, and f(0.5) = 0.25.

3. Relation: {(a,b) | a and b are integers, and a + b = 5}

This is a relation between the set of integers and itself. It establishes a connection between pairs of integers that add up to 5.

4. Function: g(x) = sin(x)

This is a function that maps each real number x to its sine, resulting in a range between -1 and 1. For example, g(π/2) = 1, g(0) = 0, and g(π) = 0.

5. Relation: {(x,y) | x and y are real numbers, and y = x²}

This is a relation between the set of real numbers and itself. It establishes a connection between pairs of real numbers where the second element is the square of the first.

6. Function: h(x) = |x|

This is a function that maps each real number x to its absolute value, resulting in a range of non-negative real numbers. For example, h(-2) = 2, h(3) = 3, and h(0) = 0.

Note that not all relations are functions, but all functions are relations. Functions have a unique output for each input, while relations may have multiple outputs for a single input.

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Properties of Real Functions

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The properties of real functions are as follows - 

1. Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. The domain and range of a real function are subsets of the real numbers.

Example: The function f(x) = x² has a domain of all real numbers and a range of non-negative real numbers (i.e., {y | y ≥ 0}).

2. Continuity

A function is continuous if it does not have any breaks or jumps in its graph. In other words, a function is continuous if it can be drawn without lifting the pencil from the paper. Continuous functions have many useful properties, such as the intermediate value theorem and the extreme value theorem.

Example: The function f(x) = sin(x) is continuous for all real numbers.

3. Differentiability

A function is differentiable if it has a derivative at every point in its domain. The derivative of a function measures its rate of change at each point, and it can be used to find the maximum and minimum values of the function. Differentiable functions have many useful properties, such as the mean value theorem and Rolle's theorem.

Example: The function f(x) = x^2 is differentiable for all real numbers, and its derivative is f'(x) = 2x.

4. Monotonicity

A function is monotonic if it always increases or always decreases as its input increases. Monotonic functions have many useful properties, such as the inverse function theorem and the intermediate value theorem.

Example: The function f(x) = e^x is strictly increasing for all real numbers.

5. Periodicity

A function is periodic if it repeats itself over a certain interval. Periodic functions are commonly used to model phenomena such as wave motion, sound propagation, and the behaviour of light.

Example: The function f(x) = sin(x) is periodic with a period of 2π.

6. Symmetry

A function is symmetric if its graph is unchanged when reflected across a certain line or point. Symmetric functions have many useful properties, such as the odd/even functions theorem and the intermediate value theorem.

Example: The function f(x) = x^3 is an odd function, meaning that f(-x) = -f(x) for all x.

7. Invertibility

A function is invertible if it has a unique inverse function. The inverse function of a function undoes the effects of the original function, allowing for the solution of equations and the manipulation of functions.

Example: The function f(x) = e^x is invertible, and its inverse function is f^-1(x) = ln(x).

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Operations on Real Functions

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The operations that can be performed on real functions are - 

1. Addition and subtraction:

Given two functions f(x) and g(x), their sum or difference can be defined as (f+g)(x) = f(x) ± g(x).

For example, if f(x) = 2x and g(x) = x^2, then (f+g)(x) = 2x + x^2 and (f-g)(x) = 2x - x^2.

2. Multiplication:

Given two functions f(x) and g(x), their product can be defined as (fg)(x) = f(x) * g(x).

For example, if f(x) = 2x and g(x) = x^2, then (fg)(x) = 2x * x^2 = 2x^3.

3. Composition:

Given two functions f(x) and g(x), their composition can be defined as (f ◦ g)(x) = f(g(x)).

For example, if f(x) = x^2 and g(x) = 2x-1, then (f ◦ g)(x) = f(g(x)) = (2x-1)^2.

4. Inverse:

Given a function f(x), its inverse function can be defined as f^-1(x), such that f(f^-1(x)) = f^-1(f(x)) = x.

For example, if f(x) = 2x, then f^-1(x) = x/2.

5. Differentiation:

Given a function f(x), its derivative f'(x) can be defined as the rate of change of the function with respect to x.

For example, if f(x) = x^2, then f'(x) = 2x.

6. Integration:

Given a function f(x), its integral F(x) can be defined as the area under the curve of the function with respect to x.

For example, if f(x) = x^2, then F(x) = (1/3)x^3 + C, where C is the constant of integration.

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Things to Remember

1. A real function is a mathematical object that assigns a unique real number to each input in its domain.
2. The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs.
3. A function can be represented graphically as a curve on a coordinate plane, with the input values on the x-axis and the output values on the y-axis.
4. Real functions can be classified as continuous or discontinuous based on their behavior at certain points in their domain.
5. Real functions can have important properties such as being monotonic (always increasing or always decreasing) or having local extrema (maximum or minimum values within a certain range of inputs).