Function operations and composition are powerful tools in algebra. They allow us to combine and manipulate functions in various ways, creating new relationships between inputs and outputs. These techniques are essential for modeling complex real-world scenarios and solving advanced mathematical problems.
Understanding how to add, subtract, multiply, and divide functions, as well as compose them, opens up a world of possibilities. These skills help us break down complicated systems into simpler components, making it easier to analyze and solve problems across many fields of study.
Function Operations and Composition
Combining functions algebraically
Addition combines two functions f and g by adding their outputs for each input value, denoted as (f+g)(x)=f(x)+g(x)
Example: If f(x)=2x and g(x)=x2, then (f+g)(x)=2x+x2
Subtraction combines two functions f and g by subtracting their outputs for each input value, denoted as (f−g)(x)=f(x)−g(x)
Example: If f(x)=3x−1 and g(x)=x+2, then (f−g)(x)=(3x−1)−(x+2)=2x−3
Multiplication combines two functions f and g by multiplying their outputs for each input value, denoted as (f⋅g)(x)=f(x)⋅g(x)
Example: If f(x)=2x and g(x)=x−1, then (f⋅g)(x)=2x(x−1)=2x2−2x
Division combines two functions f and g by dividing their outputs for each input value, denoted as (gf)(x)=g(x)f(x), where g(x)=0 to avoid division by zero
Example: If f(x)=x2 and g(x)=x, then (gf)(x)=xx2=x, for x=0
Creation of composite functions
involves applying one function to the output of another, denoted as (f∘g)(x)=f(g(x)), where the output of function g becomes the input for function f
Example: If f(x)=x and g(x)=x2+1, then (f∘g)(x)=x2+1
To evaluate a , first evaluate the inner function g(x), then use the result as the input for the outer function f
Example: To evaluate (f∘g)(3) using the functions from the previous example:
Evaluate g(3)=32+1=10
Evaluate f(10)=10≈3.16
Therefore, (f∘g)(3)≈3.16
Nested functions are a form of function composition where one function is contained within another (e.g., f(g(x)))
Domain of composite functions
The (f∘g)(x) consists of all x values such that x is in the domain of g and g(x) is in the domain of f
Example: If f(x)=x and g(x)=x−2, then:
Domain of g: all real numbers
Domain of f: x≥0
For g(x) to be in the domain of f, we need x−2≥0, or x≥2
Therefore, the domain of (f∘g)(x) is x≥2
To find the domain of a composite function, determine the values of x that satisfy the domain conditions for both the inner and outer functions
Decomposition of composite functions
Decomposing a composite function (f∘g)(x) involves identifying the outer function f and the inner function g, then expressing each component function separately
Example: If (f∘g)(x)=2x+1, then:
Inner function: g(x)=2x+1
Outer function: f(x)=x
To , rewrite it as f(g(x)) and identify the expressions that correspond to the inner and outer functions
Applications of function composition
Function composition can model real-world situations where the output of one process becomes the input for another
Example: Converting temperature from Fahrenheit to Kelvin
First, convert Fahrenheit to Celsius: C(F)=95(F−32)
Then, convert Celsius to Kelvin: K(C)=C+273.15
The composite function is K(C(F))=95(F−32)+273.15
Other applications include:
Calculating the cost of production and the resulting profit ( composed with revenue function)
Determining the height of an object thrown upward at a given time (velocity function composed with position function)
Modeling population growth or decay over time (growth rate function composed with population function)
Function Notation and Related Concepts
Function notation expresses a function as f(x), where f is the function name and x is the input variable
Inverse functions "undo" the effect of a function, denoted as f−1(x), and exist only for one-to-one functions
One-to-one functions map each element of the domain to a unique element in the range
Function graphs visually represent the relationship between input and output values
Piecewise functions are defined by different expressions over different intervals of the domain
Key Terms to Review (7)
Combining functions: Combining functions involves creating a new function by applying one function to the results of another. This is often done through operations such as addition, subtraction, multiplication, division, and composition.
Commutative: A property of binary operations where the order of the operands does not change the result. For functions, it implies $f(g(x)) = g(f(x))$ for all $x$ in the domain.
Composite function: A composite function is a function formed by applying one function to the results of another. It is denoted as $(f \circ g)(x) = f(g(x))$.
Composition of functions: Composition of functions is the process of applying one function to the results of another. If $f(x)$ and $g(x)$ are two functions, the composition is written as $(f \circ g)(x) = f(g(x))$.
Cost function: A cost function represents the total cost of producing a given quantity of output, typically expressed as a mathematical equation. It is often used in economics and business to model the costs associated with production.
Decompose a composite function: Decomposing a composite function involves breaking down a complex function into simpler functions whose composition yields the original function. This process helps in understanding and simplifying the behavior of composite functions.
Domain of a composite function: The domain of a composite function $f(g(x))$ is the set of all real numbers $x$ such that $x$ lies in the domain of $g(x)$ and $g(x)$ lies in the domain of $f(x)$. It ensures that both functions are defined for the input values used.