6.05 Function Composition

Important

Paying attention to the units that both functions, f(x) and g(x), are measured in will help you figure out what the new function, f(g(x)), will represent. If f(x) is measured in dollars/bagel and g(x) is measured in bagels/hour then the composition of f(g(x)) is taking a function measured in bagels/hour and substituting that into a function being measured in dollars/bagel This means the composition f(g(x)) will be measured in dollarshour

Sum It Up

- The notation used to indicate the composition of functions f(x) and g(x) may look like f(g(x)), (f ∘ g)(x), g(f(x)), or (g ∘ f)(x). For example, just as f(1) means substitute 1 for the variable in function f, f(g(x)) means substitute g(x) for the variable in function f. - To find the composite of a function, work from the inside out. First, evaluate the inside function. Next, substitute that output into the outside function. - Paying attention to the units that both functions, f(x) and g(x), are measured in will help you figure out what the new function, f(g(x)), will represent.

Example 1: If f(x) = x + 2 and g(x) = 2x − 5, find f[g(4)] This problem is asking you what happens when x is replaced with 4 for the function g. After this is evaluated, that number gets plugged into function f. Remember to work from the inside out.

Begin by substituting 4 into g(x) - g(x) = 2x − 5 - g(4) = 2(4) − 5 - g(4) = 8 − 5 - g(4) = 3 Now that you know that g(4) = 3, substitute the 3 into f(x). - f(x) = x + 2 - f(3) = 3 + 2 - f(3) = 5 So, f[g(4)] = 5

Function Composition

Function composition involves substituting one function into the variable in the other function. This process is the same as if you are evaluating a function for a specific value. The notation used to indicate the composition of functions f(x) and g(x) may look like f(g(x)), (f ∘ g)(x), g(f(x)), or (g ∘ f)(x). For example, just as f(1) means substitute 1 for the variable in function f, f(g(x)) means substitute g(x) for the variable in function f. Function composition is different from function multiplication. By substituting the function g(x) into the function f(x), you are evaluating f(x) at g(x). As g(x) changes for each x-value, the composition of f(g(x)) will also change because you will be evaluating f(x) at the new g(x) value each time. If you are given a specific value for g(x), then you could simply substitute that value into f(x) to solve for the composition, f(g(x)). For example, if you are given g(4) = 10 and are asked to find f(g(4)), you would substitute 10 into f(x). You are evaluating f(x) at the answer for g(4). Most often, you will need to write a new function, f(g(x)), using the original functions so that you can have a new function that will represent all x-values.

Rory knows that he burns around 400 calories per hour while jogging. He can express this with the function c(h) = 400h, where c is the calories burned from jogging and h is the number of hours jogging. Rory really wants to increase his fitness level, so he would like to start wearing ankle weights, which multiplies the number of calories burned by 1.5. This scenario can be expressed with the function t(c) = 1.5c, where t is the total calories burned. Help Rory find out how many calories he could burn in any amount of time while jogging and wearing ankle weights by finding t[c(h)].

Substitute c(h) into t(c). c(h) = 400h t(c) = 1.5c t[c(h)] = 1.5(400h) t[c(h)] = 600h Rory knows that he can substitute any number of hours into the variable h and find the number of calories burned during his exercise with ankle weights with the function t[c(h)] = 600h

Composite Functions

The term composite function refers to a process, where the output of one function is the input for the next function. There is special notation used to describe the order to find a composite function. If you have the composite function f[g(x)], g(x) is evaluated first and the output value is then used as the input value for function f(x). You may also see this written as (f ∘ g)(x). Both f[g(x)] and (f ∘ g)(x) are said aloud as "f of g of x."

Example: If f(x) = x − 4 and g(x) = 5x, find f[g(5)]

g(x) = 5x g(5) = 5(5) g(5) = 25 So, f[g(5)] becomes f(25). Then, f(x) = x − 4 f(25) = 25 − 4 f(25) = 21 So, f[g(5)] = 21