Mathematical functions are fundamental constructs that define relationships between variables in daily life, encompassing physics, economics, and biology. They can be linear, quadratic, or exponential, each with unique properties and applications. Functions are also classified by symmetry as even or odd, and by mapping properties as injective, surjective, or bijective. Understanding these functions is crucial for modeling real-world phenomena and interpreting mathematical models.
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Functions are mathematical constructs that describe the relationships between variables and have a specific input-output mapping
Functions are essential in modeling and understanding real-world situations in a precise and analytical way
Functions can be expressed in different forms, such as algebraic equations and graphical representations
Algebraic functions are constructed using algebraic operations on variables and constants, such as addition, subtraction, multiplication, division, and exponentiation
Linear Functions
Linear functions have the form f(x)=mx+b and their graphs are lines
Quadratic Functions
Quadratic functions have the form f(x)=ax^2+bx+c and their graphs are parabolas
Higher-Degree Polynomials
Higher-degree polynomial functions have the form f(x)=ax^n+bx^(n-1)+...+cx+d and their graphs have various shapes
Functions can exhibit even or odd symmetry, depending on their algebraic structure
Quadratic functions are a subset of algebraic functions with the form f(x)=ax^2+bx+c, where a, b, and c are constants and a ≠ 0
Quadratic functions are widely used in various fields to model phenomena such as projectile motion and areas of geometric shapes
The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of a
Injective functions have a one-to-one mapping between the domain and codomain, where different inputs map to different outputs
Surjective functions have a mapping where every element in the codomain is the image of at least one element in the domain
Bijective functions are both injective and surjective, establishing a one-to-one correspondence between the domain and codomain
Exponential functions have the form f(x)=a^x, where a is a positive constant and x is the exponent
Exponential functions are crucial in modeling growth and decay processes in natural and social sciences
The graph of an exponential function is a curve that can show rapid growth or decay, with the base determining the rate of change
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Definition of a mathematical function
A function pairs each input with exactly one output.
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Forms of expressing functions
Functions can be algebraic equations or graphical representations.
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Examples of algebraic function expressions
Quadratic function: f(x)=x^2, Polynomial function: g(x)=x^4+3.
