Combinations and Compositions of Functions: Apply It 1

  • 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

Understanding Composite Functions

Now that you have learned about composite functions and their domains, let’s interpret what a composite function means.

A composite function combines two functions where the output of one function becomes the input of another.If we have two functions [latex]f(x)[/latex] and [latex]g(x)[/latex], the composite function [latex](f \circ g)(x)[/latex] means we first apply [latex]g(x)[/latex]and then apply [latex]f[/latex] to the result of [latex]g(x)[/latex].

Composite functions can model real-world scenarios where a series of processes or transformations are applied sequentially.

Recall the example: Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year.Notice how we have just defined two relationships:

  • The cost depends on the temperature
  • the temperature depends on the day

Using descriptive variables, we can notate these two functions.

  • The function [latex]C\left(T\right)[/latex] gives the cost [latex]C[/latex] of heating a house for a given average daily temperature in [latex]T[/latex] degrees Celsius.
  • The function [latex]T\left(d\right)[/latex] gives the average daily temperature on day [latex]d[/latex] of the year.

For any given day, [latex]\text{Cost}=C\left(T\left(d\right)\right)[/latex] means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature [latex]T\left(d\right)[/latex].

For example, we could evaluate [latex]T\left(5\right)[/latex] to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write [latex]C\left(T\left(5\right)\right)[/latex].

Explanation of C(T(5)), which is the cost for the temperature and T(5) is the temperature on day 5.

The function [latex]c\left(s\right)[/latex] gives the number of calories burned completing [latex]s[/latex] sit-ups, and [latex]s\left(t\right)[/latex] gives the number of sit-ups a person can complete in [latex]t[/latex] minutes.Interpret [latex]c(s(3))[/latex].


To interpret [latex]c(s(3))[/latex], follow these steps:

  • Identify [latex]s(3)[/latex]: This represents the number of sit-ups a person can complete in [latex]3[/latex] minutes. So, [latex]s(3)[/latex] tells us how many sit-ups are done in [latex]3[/latex] minutes.
  • Apply [latex]c[/latex] to [latex]s(3)[/latex]: Once you know the number of sit-ups completed in [latex]3[/latex] minutes (which is [latex]s(3)[/latex], you use the function [latex]c[/latex] to determine how many calories are burned from doing that number of sit-ups.

Thus, [latex]c(s(3))[/latex] represents the number of calories burned from the number of sit-ups that can be completed in [latex]3[/latex] minutes.

In other words, you first calculate how many sit-ups are completed in [latex]3[/latex] minutes, and then determine the calories burned from that amount of exercise.