Problem 21 - Algebra 1: Concepts and Skills

Question

STATING INVERSES State the inverse operation. Divide by 7

Step-by-Step Solution

Answer

The inverse operation of dividing by 7 is multiplying by 7.

Step 1: Understand the original operation

The original operation given in the exercise is dividing by 7. This operation can be represented as \( f(x) = \frac{x}{7} \) where \( f(x) \) is a function that divides an input \( x \) by 7.

Step 2: Identify the inverse operation

In mathematics, the inverse operation of division is multiplication. Therefore, the inverse operation of dividing by 7 is multiplying by 7. This can be represented as \( f^{-1}(x) = 7x \) where \( f^{-1}(x) \) is the function that multiplies an input \( x \) by 7.

Key Concepts

Division

Division is one of the basic arithmetic operations where we split a number into equal parts. Think of it as sharing something equally among a certain number of people. For example, if you have 14 cookies and you want to divide them among 7 friends, each friend would get 2 cookies because 14 divided by 7 equals 2.
In mathematical terms, when we divide, we are determining how many times one number, called the divisor (in our example, 7), fits into another number, called the dividend (in our example, 14). The result is the quotient (2 in this case).
In the context of functions, division can be expressed as a function like this: for a given input number, our function gives the result of dividing this number by a fixed divisor, say 7:

• Function notation: \( f(x) = \frac{x}{7} \)
• This tells us to divide any input \( x \) by 7.

Multiplication

Multiplication is the inverse operation of division. This means that if you reverse a division operation, you use multiplication.
Multiplication combines equal groups of things; it’s essentially repeated addition. For example, if you multiply 3 by 4, it’s like saying 4 plus 4 plus 4, which equals 12.
When we want to find the inverse of a division operation that divides by a number, like 7, we multiply by that number.

• Inverse operation notation: \( f^{-1}(x) = 7x \)
• This tells us to take any input \( x \) and multiply it by 7.

Think of multiplication as the process that helps us undo a division. If we originally divided by 7, multiplying by 7 would take us back to our starting point.

Functions

Functions are the foundations of mathematics that map an input to an output based on a specific rule. They are like machines where you put something in, perform an operation, and get something out.
For example, with the function \( f(x) = \frac{x}{7} \), you input a number \( x \) and the function outputs the number divided by 7.
To find the opposite action, which is known as the inverse function, you perform the reverse operation, which in our case is multiplication. This is denoted as \( f^{-1}(x) = 7x \), taking the result from the function \( f(x) \) and returning to the original input by multiplying by 7.

• Functions help us model real-world scenarios and relate quantities.
• Inverse functions allow us to reverse operations.

Understanding functions and their inverses in division and multiplication showcases the balance and interrelationship between operations in math.