Combinations and Compositions of Functions: Fresh Take

  • Use algebraic operations to combine functions and create new expressions
  • Build a new function by combining two or more functions together
  • Calculate the output for composite functions for given values and determine the set of inputs that work for these functions
  • Break down a composite function into the original functions that were combined to make it

Combining Functions Using Algebraic Operations

The Main Idea

  • Function Addition:
    • [latex](f + g)(x) = f(x) + g(x)[/latex]
  • Function Subtraction:
    • [latex](f - g)(x) = f(x) - g(x)[/latex]
  • Function Multiplication:
    • [latex](f \cdot g)(x) = f(x) \cdot g(x)[/latex]
  • Function Division:
    • [latex]\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}[/latex], where [latex]g(x) \neq 0[/latex]
  • Domain Considerations:
    • Resulting function’s domain may be restricted, especially for division
Find and simplify the functions [latex]\left(f\cdot{g}\right)\left(x\right)[/latex] and [latex]\left(f-g\right)\left(x\right)[/latex].

[latex]f\left(x\right)=x - 1\text{ and }g\left(x\right)={x}^{2}-1[/latex]

Are they the same function?

Create a Function by Composition of Functions

The Main Idea

  • Definition:
    • [latex](f \circ g)(x) = f(g(x))[/latex]
    • Read as “f composed with g of x” or “f of g of x”
  • Order Matters:
    • Generally, [latex](f \circ g)(x) \neq (g \circ f)(x)[/latex]
  • Domain Considerations:
    • Domain of [latex]f \circ g[/latex]: All [latex]x[/latex] in domain of [latex]g[/latex] where [latex]g(x)[/latex] is in domain of [latex]f[/latex]
  • Evaluation Process:
    • Work from innermost function outward
  • Not Multiplication:
    • [latex](f \circ g)(x) \neq (f \cdot g)(x)[/latex]
The gravitational force on a planet a distance [latex]r[/latex] from the sun is given by the function [latex]G\left(r\right)[/latex]. The acceleration of a planet subjected to any force [latex]F[/latex] is given by the function [latex]a\left(F\right)[/latex]. Form a meaningful composition of these two functions, and explain what it means.

Using the functions provided, find [latex]f\left(g\left(x\right)\right)[/latex] and [latex]g\left(f\left(x\right)\right)[/latex].
[latex]f\left(x\right)=2x+1\\g\left(x\right)=3-x[/latex]

You can view the transcript for “Composite Functions” here (opens in new window).

You can view the transcript for “Ex 4: Domain of a Composite Function” here (opens in new window).

Evaluating Composite Functions

The Main Idea

  • General Approach:
    • Work from inside to outside
    • Evaluate inner function first, use result as input for outer function
  • Using Tables:
    • Read input/output values directly from table entries
    • [latex](f \circ g)(x) = f(g(x))[/latex]: Look up [latex]g(x)[/latex], then use result to find [latex]f[/latex] value
  • Using Graphs:
    • Read input/output values from [latex]x[/latex] and [latex]y[/latex] axes
    • Follow points across graphs for composite functions
  • Using Formulas:
    • Substitute expression for inner function into outer function
    • Simplify to get formula for composite function
  • Domain Considerations:
    • Check domains of both inner and outer functions
    • Ensure composition is valid for given input
Using the table below, evaluate [latex]f\left(g\left(1\right)\right)[/latex] and [latex]g\left(f\left(4\right)\right)[/latex].

[latex]x[/latex] [latex]f\left(x\right)[/latex] [latex]g\left(x\right)[/latex]
1 6 3
2 8 5
3 3 2
4 1 7

Using the graphs below, evaluate [latex]g\left(f\left(2\right)\right)[/latex].Two graphs of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.

Given [latex]f\left(t\right)={t}^{2}-{t}[/latex] and [latex]h\left(x\right)=3x+2[/latex], evaluate [latex]f\left(h\left(1\right)\right)[/latex].

Given [latex]f\left(t\right)={t}^{2}-t[/latex] and [latex]h\left(x\right)=3x+2[/latex], evaluatea.  [latex]h\left(f\left(2\right)\right)[/latex]b.  [latex]h\left(f\left(-2\right)\right)[/latex]

You can view the transcript for “Ex: Evaluate Composite Functions from Graphs” here (opens in new window).

You can view the transcript for “Ex 1: Composite Function Values” here (opens in new window).

Decomposing a Composite Function

The Main Idea

  • Definition:
    • Breaking down a complex function into simpler component functions
    • [latex]f(x) = g(h(x))[/latex], where [latex]g[/latex] and [latex]h[/latex] are simpler functions
  • Multiple Solutions:
    • There’s often more than one way to decompose a function
    • Choose the decomposition that seems most useful or intuitive
  • Process:
    • Identify a “function inside a function” in the original expression
    • Define inner function [latex]h(x)[/latex] and outer function [latex]g(x)[/latex]
  • Verification:
    • Recompose the functions to check if [latex]g(h(x))[/latex] equals the original function
  • Applications:
    • Simplifying complex functions
    • Understanding function structure
    • Solving certain types of equations
Write [latex]f\left(x\right)=\dfrac{4}{3-\sqrt{4+{x}^{2}}}[/latex] as the composition of two functions.

You can view the transcript for “Ex: Decompose Functions” here (opens in new window).