Problem 48 - Algebra and Trigonometry

Question

Find the inverse of \(f(x)=3 x-4\).

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=3x-4\) is \(f^{-1}(x)=(x+4)/3\)

Step 1: Replace \(f(x)\) with \(y\)

To make the operations easier, replace \(f(x)\) with \(y\). So, instead of \(f(x)=3x-4\), the function becomes \(y=3x-4\)

Step 2: Swap \(x\) and \(y\)

Now, swap the places of \(x\) and \(y\). The equation then becomes \(x=3y-4\)

Step 3: Solve for \(y\)

To find the inverse function, solve this equation for \(y\). First, isolate the term with \(y\) by adding 4 to both sides. This results in the equation \(x+4=3y\). Next, divide both sides by 3 to solve for \(y\). The final inverse function is \(y=(x+4)/3\)

Step 4: Replace \(y\) with \(f^{-1}(x)\)

Now that you have solved for \(y\), replace it with \(f^{-1}(x)\). Therefore, the inverse function is \(f^{-1}(x)=(x+4)/3\).

Key Concepts

Function Notation

Function notation is a compact and convenient way to represent functions in mathematics. It helps us identify and work with functions more easily by using symbols like \( f(x) \). In the context of a function, \( f \) is the name of the function, and \( x \) is the variable or input. For example, \( f(x) = 3x - 4 \) means that \( f \) outputs a value that is 3 times the input \( x \), minus 4.

• When you see \( f(x) \), remember it's just a name, not a multiplication of \( f \) and \( x \).
• Function notation allows us to evaluate a function by replacing \( x \) with a specific number. For example, \( f(2) \) would mean substituting 2 into the function, so \( f(2) = 3(2) - 4 \).

In inverse functions, we use another notation, \( f^{-1}(x) \), to denote the inverse of \( f(x) \). This notation tells us that \( f^{-1} \) will reverse the effect of \( f \). For instance, if \( f(x) = 3x - 4 \), then \( f^{-1}(x) = \frac{x+4}{3} \) reverses that, transforming the output back into the original input.

Solving Equations

Solving equations is about finding the value of a variable that makes the equation true. When finding the inverse of a function, equation solving is key, because you interchange and solve for the dependent variable to find its inverse.To solve the equation \( y = 3x - 4 \) for its inverse:

• We start by swapping \( x \) and \( y \). This means setting \( x = 3y - 4 \), which means that what was the output is now the input.
• Next, we isolate \( y \) by reversing the operations done to \( x \). To do this, first add 4 to both sides, giving us \( x + 4 = 3y \).
• Then, divide both sides by 3 to solve for \( y \), resulting in \( y = \frac{x + 4}{3} \), thus yielding the inverse function.

Solving equations is a fundamental skill that allows us to understand relationships between variables in inverse functions and beyond.

Algebraic Manipulation

Algebraic manipulation refers to the process of re-arranging and simplifying equations to find solutions. It involves performing operations like addition, subtraction, multiplication, and division to transform equations into a more workable form.Here’s how algebraic manipulation was used in finding the inverse of the function \( y = 3x - 4 \):

• Once you swap \( x \) and \( y \) as in \( x = 3y - 4 \), the first step is to add 4 to both sides to undo the subtraction of 4. This manipulation results in \( x + 4 = 3y \).
• The next manipulation involves dividing both sides by 3 to solve for \( y \). This step transforms the equation into \( y = \frac{x + 4}{3} \).

This manipulation is not just trial and error; it’s a logical method that uses inverse operations to gradually simplify and solve the equation. Mastering these techniques can greatly enhance your ability to solve more complex algebraic problems effectively.