Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=3 x+5 $$

Step-by-Step Solution

Answer

The inverse of the function is \( f^{-1}(x) = \frac{x - 5}{3} \).

Step 1: Understand the Function

We are given the function \( f(x) = 3x + 5 \). To find its inverse, we need to rearrange the equation so that we express \( x \) in terms of \( y \), where \( y = f(x) \).

Step 2: Replace \( f(x) \) with \( y \)

First, rewrite the equation with \( y \) in place of \( f(x) \). This gives us: \( y = 3x + 5 \).

Step 3: Solve for \( x \)

To find the inverse, solve the equation \( y = 3x + 5 \) for \( x \). Start by subtracting 5 from both sides to isolate the term with \( x \): \( y - 5 = 3x \).

Step 4: Divide by 3

Now divide each side by 3 to solve for \( x \): \( x = \frac{y - 5}{3} \).

Step 5: Write the Inverse Function

Now that we have expressed \( x \) in terms of \( y \), we replace \( y \) with \( f^{-1}(x) \) to denote the inverse function: \( f^{-1}(x) = \frac{x - 5}{3} \).

Key Concepts

Linear Functions

Linear functions are mathematical expressions that create a straight line when graphed. They are defined as functions in the form of \( y = mx + b \), where \( m \) represents the slope and \( b \) stands for the y-intercept. In our given function \( f(x) = 3x + 5 \), we have:

• Slope \( m = 3 \)
• Y-intercept \( b = 5 \)

This means that for every unit increase in \( x \), \( y \) increases by 3. Linear functions are straightforward because they don't involve complicated operations like squares or roots.
Their simplicity makes them ideal for beginners starting to learn about functions or graphing.
Linear functions model many real-life situations, like calculating distance over time at a constant speed or converting temperatures. Understanding linear functions is essential for studying more complex types of functions later.

Solving Equations

To find the inverse of a linear function, you need to solve an equation. Solving equations means isolating one variable on one side of the equation so you can express it in terms of another variable. In the step-by-step solution of our function \( y = 3x + 5 \), solving for \( x \) involves:

• Subtracting 5 from both sides: \( y - 5 = 3x \)
• Dividing both sides by 3: \( x = \frac{y - 5}{3} \)

This step-by-step process removes coefficients and constants to solve for the variable of interest.
These skills are vital in algebra, allowing students to manipulate equations to find unknown values.
Being adept at solving equations prepares students for more advanced math topics, such as calculus and differential equations.
In our context, mastering this skill enables the expression of linear function inverses, enhancing comprehension of function behavior.

Function Notation

Function notation provides a way to express mathematical functions concisely and precisely, using symbols like \( f(x) \) to denote functions. The notation is used to highlight inputs and outputs in functions. In the original problem, \( f(x) = 3x + 5 \), \( f(x) \) means the function's output when \( x \) is the input.
When dealing with inverse functions, such as \( f^{-1}(x) \), we denote the function that reverses the effect of \( f(x) \). In summary:

• \( f(x) \) represents the original function
• \( f^{-1}(x) \) represents the inverse function

This notation helps in straightforwardly understanding changes, relationships, and solving equations involving functions.
Function notation is an essential part of mathematical language, enabling effective communication of complex ideas in a simple manner.
For students, mastering function notation simplifies learning other math topics and enhances problem-solving strategies.