MM - Operations and Composition of Functions Lesson

Operations and Composition of Functions

We have discussed operations and composition of functions in a previous unit, but now we will review and expand some on our knowledge. Operations can be done with functions just like with expressions. Here is the notation we will use. The examples will use the functions f(x)= 2x + 3 and g(x) = x² - x.

Operation	Notation	Example
Addition	$LaTeX: \left(f+g\right)\left(x\right)\:or\:\:f\left(x\right)+g\left(x\right)$ $\left(f+g\right)\left(x\right)\:or\:\:f\left(x\right)+g\left(x\right)$	$LaTeX: (2x+3)+\left(x^2-x\right)=x^2+x+3$ $(2x+3)+\left(x^2-x\right)=x^2+x+3$
Subtraction	$LaTeX: \left(f-g\right)\left(x\right)\:or\:\:f\left(x\right)-g\left(x\right)$ $\left(f-g\right)\left(x\right)\:or\:\:f\left(x\right)-g\left(x\right)$	$LaTeX: (2x+3)-\left(x^2-x\right)=-x^2+x+3$ $(2x+3)-\left(x^2-x\right)=-x^2+x+3$
Multiplication	$LaTeX: \left(f\cdot g\right)\left(x\right)\:or\:\:f\left(x\right)\cdot g\left(x\right)$ $\left(f\cdot g\right)\left(x\right)\:or\:\:f\left(x\right)\cdot g\left(x\right)$	$LaTeX: (2x+3)\cdot\left(x^2-x\right)=2x^3+x^2-3$ $(2x+3)\cdot\left(x^2-x\right)=2x^3+x^2-3$
Division	$LaTeX: \left(f/g\right)\left(x\right)\:or\:\:f\left(x\right)/g\left(x\right)$ $\left(f/g\right)\left(x\right)\:or\:\:f\left(x\right)/g\left(x\right)$	$LaTeX: \left(2x+3\right)/\left(x^2-x\right)$ $\left(2x+3\right)/\left(x^2-x\right)$ , cannot be simplified. Also, in division remember that denominators cannot be zero. $LaTeX: g\left(x\right)\ne0$ $g\left(x\right)\ne0$

These "combinations" of functions use the same properties you learned in previous modules. You can find the domain of each by determining what values can be used for x, the same as in previous modules. Always simplify your answers, where possible.

Watch the following videos that will review and explore operations of functions.

It's now time for us to explore and practice working Operations with Functions.

*Please note in the following learning object, answer 2 should say -2x^2 not -2x^3.

Composite Functions

The next thing to look at is composite functions.

Composition of functions is when we are given two functions, their composite (combined function) uses the output from one function as the input for the other function. We will see the common notation of f(x) and g(x), representing the two different functions.

Notation for composite functions is

equation image indicator $LaTeX: \left(f\circ g\right)\left(x\right)\:or\:f\left(g\left(x\right)\right)$ $\left(f\circ g\right)\left(x\right)\:or\:f\left(g\left(x\right)\right)$

The second notation allows us to see what the input is and what the output is.

In f(g(x)), the output for the function g(x) is used as the input for f(x) .
In g(f(x)), the output for the function f(x) is used as the input for g(x).

The domain of composite functions is determined by the domains of the original functions, not the resulting function.

The purpose of composing functions is often to evaluate the result for a specific number.

For example, if f(x) = 2x + 3 and g(x) = x² - x , we might want to find equation image indicator $LaTeX: \left(f\circ g\right)\left(3\right)\:or\:f\left(g\left(3\right)\right)$ $\left(f\circ g\right)\left(3\right)\:or\:f\left(g\left(3\right)\right)$ .

To do this, we don't have to find f(g(x)).We can simply follow order of operations, doing what is in the ( ) first.

So we find g(3) by putting 3 in for x in the g(x) function.

g(3) = 3² - 3 = 9 - 3 = 6

Now we take g(3) which is 6 and put it in for x in the f(x) function.

f(g(3)) = f(6) = 2(6) + 3 = 15

Watch the following videos that will review and explore composition of functions.

Let's see how this is used.

Aisha made a chart of the experimental data for her science project and showed it to her science teacher.

The teacher was complimentary of Aisha's work but suggested that, for a science project, it would be better to list the temperature data in degrees Celsius rather than degrees Fahrenheit.

a. Aisha found the formula for converting from degrees Fahrenheit to degrees Celsius- equation image indicator $LaTeX: C=\frac{5}{9}\left(F-32\right)$ $C=\frac{5}{9}\left(F-32\right)$ .

Use this formula to convert freezing (32°F) and boiling (212°F) to degrees Celsius.

Solution: Freezing is 0 degrees Celsius and boiling is 100 degrees Celsius

b. Later Aisha found a scientific journal article related to her project and planned to use information from the article on her poster for the school science fair.

The article included temperature data in degrees Kelvin.

Aisha talked to her science teacher again, and they concluded that she should convert her temperature data again this time to degrees Kelvin.

The formula for converting degrees Celsius to degrees Kelvin is K = C + 273. Use this formula and the results of part a to express freezing and boiling in degrees Kelvin.

Solution: Freezing is 273 degrees Kelvin and boiling is 373 degrees Kelvin

c. Use the formulas from part a and part b to convert the following to °K- 238°F, 5000°F .

Solution: -238 degrees F is 123 K and 5000 degrees F is 3302 degrees K

In converting from degrees Fahrenheit to degrees Kelvin, you used two functions, the function for converting from degrees Fahrenheit to degrees Celsius and the function for converting from degrees Celsius to degrees Kelvin, and a procedure that is the key idea in the composition of functions.

We now explore how the temperature conversions from Item 1, part c, provide an example of a composite function.

d. The definition of composition of functions indicates that we start with a value, x, and first use this value as input to the function g.

In our temperature conversion, we started with a temperature in degrees Fahrenheit and used the formula to convert to degrees Celsius, so the function g should convert from Fahrenheit to Celsius- equation image indicator $LaTeX: g\left(x\right)=\frac{5}{9}\left(x-32\right)$ $g\left(x\right)=\frac{5}{9}\left(x-32\right)$ .

What is the meaning of x and what is the meaning of g(x) when we use this notation?

Solution: Here "x" is a temperature in degrees Fahrenheit and "g(x)" is the corresponding temperature is degrees Celsius

e. In converting temperature from degrees Fahrenheit to degrees Kelvin, the second step is converting a Celsius temperature to a Kelvin temperature.

The function f should give us this conversion; thus, f(x) = x + 273.

What is the meaning of x and what is the meaning of f (x) when we use this notation?

Solution: Here "x" is a temperature in degrees Celsius and "f(x)" is the corresponding temperature in degrees Kelvin

f. Perform the composition equation image indicator $LaTeX: \left(f\circ g\right)\left(45\right)\:or\:f\left(g\left(45\right)\right)$ $\left(f\circ g\right)\left(45\right)\:or\:f\left(g\left(45\right)\right)$ What is the meaning of this number?

Solution: $LaTeX: g\left(45\right)=\frac{5}{9}\left(45-32\right)=\frac{5}{9}\left(13\right)=\frac{65}{9}=7.2 \\ f\left(g\left(45\right)\right)=f\left(\frac{65}{9}\right)+273=\frac{2522}{9}\approx280.2 \\ \text{So 280.2 is the temperature in degrees K that corresponds to 45 degrees F}$ $g\left(45\right)=\frac{5}{9}\left(45-32\right)=\frac{5}{9}\left(13\right)=\frac{65}{9}=7.2 \\ f\left(g\left(45\right)\right)=f\left(\frac{65}{9}\right)+273=\frac{2522}{9}\approx280.2 \\ \text{So 280.2 is the temperature in degrees K that corresponds to 45 degrees F}$

g. Perform the composition equation image indicator $LaTeX: \left(f\circ g\right)\left(x\right)\:or\:f\left(g\left(x\right)\right)$ $\left(f\circ g\right)\left(x\right)\:or\:f\left(g\left(x\right)\right)$ , and simplify the result. What is the meaning of x and what is the meaning of $LaTeX: \left(f\circ g\right)\left(x\right)$ $\left(f\circ g\right)\left(x\right)$ ?

Solution: Here "x" is a temperature in degrees Fahrenheit and " $LaTeX: \left(f\circ g\right)\left(x\right)$ $\left(f\circ g\right)\left(x\right)$ " is the corresponding temperature in degrees Kelvin.

h. Perform the composition equation image indicator $LaTeX: \left(f\circ g\right)\left(45\right)\:or\:f\left(g\left(45\right)\right)$ $\left(f\circ g\right)\left(45\right)\:or\:f\left(g\left(45\right)\right)$ using the formula from part d. Does your answer agree with your calculation from part c?

Solution: YES

i. Perform the composition equation image indicator $LaTeX: \left(g\circ f\right)\left(x\right)\:or\:g\left(f\left(x\right)\right)$ $\left(g\circ f\right)\left(x\right)\:or\:g\left(f\left(x\right)\right)$ , and simplify the result. What is the meaning of x? What meaning, if any, relative to temperature conversion can be associated with the value of equation image indicator $LaTeX: \left(g\circ f\right)\left(x\right)$ $\left(g\circ f\right)\left(x\right)$ ?

Solution: Here "x" is a temperature in degrees Celsius. We cannot associate a meaning to " $LaTeX: \left(g\circ f\right)\left(x\right)$ $\left(g\circ f\right)\left(x\right)$ " relative to temperature conversion since "f(x)" is a temperature in degrees Kelvin, but an input to the function g should be a temperature in degrees Fahrenheit.

It's now time for us to explore and practice working Composition of Functions.

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