Problem 10 - College Algebra with Modeling and Visualization

Question

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Multiply \(x\) by \(-2\) and add 3.

Step-by-Step Solution

Answer

Inverse: Subtract 3 from \(x\) and divide by \(-2\). Symbolically, inverse is \(f^{-1}(x) = \frac{x - 3}{-2}\).

Step 1: Understand the Given Statement

The statement is 'Multiply \(x\) by \(-2\) and add 3'. This can be interpreted mathematically as the operation or function \(f(x) = -2x + 3\).

Step 2: Verbalize the Inverse Operation

To reverse the operation, we need to start from the end and reverse each step. Therefore, the inverse will be: 'Subtract 3 from the result, and then divide by \(-2\).' This will effectively cancel out the addition of 3 and multiplication by \(-2\).

Step 3: Derive the Symbolic Expression for the Inverse

Given the function \(f(x) = -2x + 3\), we denote \(y = f(x)\). Thus, the inverse operation should solve \(y = -2x + 3\) for \(x\). So, starting from \(y\):1. Subtract 3 from both sides: \(y - 3 = -2x\).2. Divide by \(-2\): \(x = \frac{y - 3}{-2}\).Thus, the inverse function can be written as \(f^{-1}(y) = \frac{y - 3}{-2}\).

Step 4: Symbolically Express the Inverse

We have already found that the inverse function is \(f^{-1}(x) = \frac{x - 3}{-2}\). This inversely corresponds to what we verbalized earlier.

Key Concepts

Function Notation

Function notation is a helpful method for representing operations in mathematics.
When dealing with a function, we use a notation like \(f(x)\) to describe how a function works on a given input, \(x\).
For example, if we have the expression \(f(x) = -2x + 3\), this implies that for any input \(x\), the function will multiply it by \(-2\) and then add 3.
Using function notation allows us to easily recognize the input and the operations performed on it.
It is crucial in performing calculations together with inverse functions, where we need to reverse operations.
Thus, with our original function \(f(x) = -2x + 3\), the challenge is to undo it, which means finding the inverse \(f^{-1}(x)\), that gives us back the initial value when applied.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations.
Expressions like \(-2x + 3\) describe a series of arithmetic steps to follow.
You engage with algebraic expressions especially when working with functions.
For instance, in our given exercise, \(-2x + 3\) is used to transform \(x\) through multiplication and addition.
The inverse of this operation can also be represented as an algebraic expression: \(\frac{x - 3}{-2}\). This describes reversing those steps.
Understanding how to manipulate these expressions is key to finding inverses.
You must be comfortable moving terms and solving equations to isolate variables.
Ultimately, algebraic expressions help form and resolve complex equations efficiently.

Operations on Functions

Operations on functions involve performing mathematical actions like addition, subtraction, multiplication, or division, often in relation to finding or using inverses.
Understanding these operations is essential for working with composite functions or solving equations.
In the given problem, we started with a function \(f(x) = -2x + 3\), which involves multiplying by \(-2\) and adding 3.
To find the inverse function, you perform opposite operations in reverse order: subtract 3 and then divide by \(-2\).
Here's a quick breakdown of the steps:

• Start with the opposite of the inner function, isolating \(x\).
• Symbolically express step-by-step until you solve for \(x\).

Recognizing that each operation on a function has an opposite is vital. This enables solving for inverses, letting you return to the initial value.
Such skills are fundamental in algebra and calculus where understanding operations thoroughly is a must.