college algebra 5.3: Combining functions

Section 5.3: Combining Functions

Overview of Topics

  • Combining functions arithmetically

  • Composing functions

  • Decomposing functions

  • Introduction to recursive graphs

Combining Functions Arithmetically

  • Mathematical operations we can perform on functions:

    • Addition: (f + g)(x) = f(x) + g(x)

    • Subtraction: (f - g)(x) = f(x) - g(x)

    • Multiplication: (fg)(x) = f(x) imes g(x)

    • Division: rac{f}{g}(x) = rac{f(x)}{g(x)}

Important Concepts

  • Notation:

    • Instead of writing out each function explicitly in operations, mathematicians often use shorthand: f + g to denote the sum of functions.

  • Domain of Combined Functions:

    • The domain of the new function (e.g., h(x) ) created by combining functions is the intersection of the domains of the original functions.

    • If either function has restrictions, those restrictions must be included in the domain of the new function.

  • Significant Restriction:

    • If a function is in the denominator, that function cannot equal zero. For example, if g(x) in the denominator, then g(x)
      eq 0 .

Example: Arithmetic Combination of Functions

Given Values

  • f(-2) = 5

  • g(-2) = -3

Calculating Values

  • For subtraction:

    • (f - g)(-2) = f(-2) - g(-2) = 5 - (-3) = 5 + 3 = 8

  • For division:

    • rac{f}{g}(-2) = rac{f(-2)}{g(-2)} = rac{5}{-3} = - rac{5}{3}

Combining Functions with Defined Expressions

Given Functions

  • f(x) = 4x^2 - 1

  • g(x) = ext{sqrt}(x)

Domain Considerations

  • The domain of f(x) is all real numbers; the domain of g(x) is [0, ∞) because of the square root restriction.

  • Hence, the combined function h(x) = f(x) + g(x) also has a domain of [0, ∞).

Operations with Defined Functions

  • Adding Functions:

    • Combined function: h(x) = 4x^2 - 1 + ext{sqrt}(x)

    • Domain: [0, ∞)

  • Multiplying Functions:

    • New function: h(x) = (4x^2 - 1) ext{sqrt}(x)

    • This can be rewritten for easier exponent addition as: (4x^2 - 1)x^{1/2}

    • Resulting in: h(x) = 4x^{ rac{5}{2}} - x^{ rac{1}{2}}

    • Domain remains [0, ∞).

Analyzing Function Graphs

Graphical Representation

  • The domains of both original functions can be graphically analyzed where:

    • f is the parabola defined by f(x) = 4x^2 - 1

    • g is the square root function.

Combining Graphically

  • Adding the graphs: The graph remains the same regarding the domain: all real numbers.

  • Dividing the two functions requires finding where g(x) is zero to avoid undefined behavior.

Composing Functions

Definition of Composition

  • Composition of functions involves substituting one function into another. It is denoted as:

    • f(g(x)) (read as “f of g of x”)

  • The correct order of operations is crucial. If we switch the order, the outcomes can differ significantly:

    • g(f(x)) is not necessarily equal to f(g(x))

    • This shows that function composition is not commutative.

Example Calculation

  • Find f(g(1)) and g(f(1)) :

    • Determine the values step-by-step:

    • For g(x) = ext{sqrt}(x) , find g(1) = 1.

    • Then plug this into f(x) resulting in f(1) = 4(1)^2 - 1 = 3 .

Composing Example Functions

  • Functions:

    • f(x) = x^2

    • g(x) = x - 3

  • Finding compositions:

    • f(g(x)) = f(x - 3) = (x - 3)^2 = x^2 - 6x + 9

    • g(f(x)) = g(x^2) = x^2 - 3

  • Demonstrating that the outcomes differ.

Recursive Functions and Graphs

  • Recursive functions iterate through a defined process, such as:

    • Given a function f(x) = x^2 ,

    • f(2) = 4

    • f(f(2)) = f(4) = 16

Recursive Notation

  • Using recursive notation, you can express iterative processes as follows:

    • f^{(n)}(x) indicates repeating the function n times.

Application of Recursive Functions in Graphics

  • Recursive functions lead to complex graphs, like the Mandelbrot set, involving fractals, which are generated through repeated iterations.

Conclusion

  • Understanding how to manipulate, compose, and decompose functions enhances our comprehension of mathematical relationships. Knowing the domain restrictions and using graphical representations aids in visualizing function behavior. Recursive functions provide insights into iterative processes, especially in advanced graphical representations.