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ACT Prep: Tutoring Solution43 chapters | 385 lessons

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Lesson Transcript

Instructor:
*Sarah Spitzig*

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree.

In this lesson, we will define functions and determine how to use specific rules for the addition, subtraction, multiplication, and division of functions.

A **function** is another way to think of an equation that has an *x* and a *y* value. We can think of *x* as the input value, or the value we plug into the equation to get the result. Likewise, we can think of *y* as the output value, or the result when we plug *x* into the equation.

For example, *y* = 3*x*, where *x* is the input and *y* is the output, in an equation. If we substituted 2 in for *x* as the input, than the output, or *y*, would be:

3(2) = 6.

Another example of an equation is *y* = -2*x* where *x* is the input and *y* is the output. If we substituted 5 in for *x* as the input, the output, or *y*, would be:

-2(5) = -10.

The only difference between an equation and a function is that instead of writing *y* as the output value, we write *f*(*x*).

The two equations *y* = 3*x* and *y* = -2*x* could be written as functions by changing the *y* to *f*(*x*).

*f*(*x*) = 3*x* is a function.

*f*'(*x*) = -2*x* is a function.

When we see *f*(*x*) = 2*x* - 3, this means that we want to find 2*x* - 3 for certain values of *x*. If I needed to find *f*(3), I would plug 3 into the equation as the input in order to get the output, so:

*f*(3) = 2*x* - 3

= 2(3) - 3

= 3

**Function operations** are rules that we follow to solve functions. There is a certain way to deal with the addition, multiplication, and division of functions.

**Function Addition Rule:**If*f*(*x*) = 3*x*- 2, and*g*(*x*) = 4*x*-1, find (*f*+*g*)(5).

First, we use the function operation rules to find (*f* + *g*)(*x*) = *f*(*x*) + *g*(*x*). Then, we use the functions we were given, *f*(*x*) and *g*(*x*), and substitute = 3*x* - 2 + 4*x* - 1, which can be simplified to = 7*x* - 3 by grouping terms.

Then, we find (*f* + *g*)(5) by plugging 5 into our new function, so:

(*f* + *g*)(*5*) = -1(5) - 3

= -8

**Function Subtraction Rule:**If*f*(*x*) = 3*x*- 2, and*g*(*x*) = 4*x*- 1, find (*f*-*g*)(5).

First, we use the operation rules to find (*f* - *g*)(*x*) = *f*(*x*) - *g*(*x*). Then, we use the functions we were given, *f*(*x*) and *g*(*x*), and substitute = 3*x* - 2 - (4*x* - 1), which can be simplified to -1*x* - 1 by grouping terms.

Then, we find (*f*- *g*)(5) by plugging 5 into our new function, so:

(*f* - *g*)(5) = -1(5) - 1

= -6

**Function Multiplication Rule:**If*f*(*x*) = 3*x*- 2, and*g*(*x*) = 4*x*- 1, find (*f***g*)(5).

First, we use the operation rules using * for multiplication to find (*f* * *g*)(*x*) = *f*(*x*) * *g*(*x*). Then, we use the functions we were given, *f*(*x*) and *g*(*x*), and substitute = (3*x* - 2)(4*x* - 1), which can be simplified to 12*x*^2 - 11*x* + 2 by multiplying 3*x* by 4*x* and 3*x* by -1 to get -12*x*^2 - 3*x*, and then multiplying -2 by 4*x* and -2 by -1 to get -8*x* + 2.

When we put this all together, we get:

-12*x*^2 - 3*x* - 8*x* + 2

Then, we need to combine like terms to get:

-12*x*^2 - 11*x* + 2

Then, we find (*f* **g*)(5) by plugging 5 into our new function, so:

(*f***g*)(5) = 12(5)^2 - 11(5) + 2

= 12(25) - 55 +2

= 247

**Function Division Rule:**If*f*(*x*) = 3*x*- 2, and*g*(*x*) = 4*x*- 1, find (*f*/*g*)(5).

First, we use the operation rule to find (*f* / *g*)(*x*) = *f*(*x*) / *g*(*x*). Then, we use the functions we were given, *f*(*x*) and *g*(*x*), and substitute = (3*x* -2) / (4*x*-1).

Then, we find (*f* / *g*)(5) by plugging 5 into our new function, so:

(*f* / *g*)(5) = (3(5) - 2) / (4(5) - 1)

= (15 - 2) / (20 - 1)

= 13 / 19

**Function notation** is another way of writing an equation using an input and an output. *y* in the equation is replaced by *f*(*x*) or *g*(*x*), or any letter (*x*) where *x* in the input and *y* is the output. Using the **function operation rules** for addition, multiplication, subtraction, and division will assist in solving these functions step by step.

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ACT Prep: Tutoring Solution43 chapters | 385 lessons

- What is a Function: Basics and Key Terms 7:57
- Inverse Functions 6:05
- Applying Function Operations Practice Problems 5:17
- How to Compose Functions 6:52
- How to Add, Subtract, Multiply and Divide Functions 6:43
- What Is Domain and Range in a Function? 8:32
- Functions: Identification, Notation & Practice Problems 9:24
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
- Polynomial Functions: Properties and Factoring 7:45
- Polynomial Functions: Exponentials and Simplifying 7:45
- Explicit Functions: Definition & Examples 7:36
- Function Operation: Definition & Overview 6:17
- Hyperbolic Functions: Properties & Applications 6:39
- Increasing Function: Definition & Example
- Parent Function in Math: Definition & Examples 7:25
- Go to ACT Math - Functions: Tutoring Solution

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