Problem 13 - College Algebra

Question

For the following exercises, find the inverse of the functions. $$ f(x)=3 x^{3}+1 $$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \sqrt[3]{\frac{x - 1}{3}} \).

Step 1: Replace f(x) with y

Begin by replacing the function notation \( f(x) \) with \( y \) to set up the equation for finding the inverse of the function:\[ y = 3x^3 + 1 \]

Step 2: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This reflects the idea that the inverse function swaps the roles of input and output:\[ x = 3y^3 + 1 \]

Step 3: Solve for y

Isolate \( y \) to express it in terms of \( x \).Subtract 1 from both sides:\[ x - 1 = 3y^3 \]Divide by 3:\[ \frac{x - 1}{3} = y^3 \]Take the cube root of both sides to solve for \( y \):\[ y = \sqrt[3]{\frac{x - 1}{3}} \]

Step 4: Write Inverse Function Notation

With \( y \) isolated, replace \( y \) with \( f^{-1}(x) \) to denote the inverse function:\[ f^{-1}(x) = \sqrt[3]{\frac{x - 1}{3}} \]

Key Concepts

Function Notation

Function notation is a way to represent functions algebraically using symbols like \( f(x) \), \( g(x) \), or any other letter. This notation provides a convenient means of displaying the relationship between inputs and outputs in a function.

• Function notation is often used because it clearly shows what variable is being used as the input.
• When you see \( f(x) = 3x^3 + 1 \), it indicates that \( f \) is a function of \( x \).

Here, \( x \) is the independent variable, and \( 3x^3 + 1 \) describes how \( y \) (or \( f(x) \)) responds to different values of \( x \). Notationally, when finding the inverse function, the roles of \( x \) and \( y \) swap.This swap changes the function from \( f(x) \) to \( f^{-1}(x) \), highlighting that the inverse function reverses the input-output roles of the original function.

Solving Equations

Solving equations involves finding the values of variables that satisfy the equation. This often requires several steps, such as isolating the variable, performing operations on both sides, and sometimes simplifying the equation.

• Simplification is key in solving equations; it makes the numbers easier to work with.
• Ensure each step maintains equal balance by doing the same operation on both sides of the equation.

In the inverse function problem, we started with the equation \( y = 3x^3 + 1 \) and swapped \( x \) and \( y \), producing \( x = 3y^3 + 1 \). Solving for \( y \) then requires a series of steps: subtraction, division, and finally taking the cube root to isolate \( y \) on one side.

Cube Root

The cube root of a number \( a \), denoted as \( \, \sqrt[3]{a} \), is a value that, when multiplied by itself three times, gives \( a \). Unlike square roots, which often concern positive numbers, both positive and negative numbers can have cube roots.

• For example, \( \, \sqrt[3]{27} = 3 \), because \( 3 \times 3 \times 3 = 27 \).
• The cube root function is the inverse of a cubing function, which we used to solve for \( y \) in the equation \( y^3 = \frac{x - 1}{3} \).

Taking the cube root here was the final step in isolating \( y \), allowing us to express \( y \) as \( \, \sqrt[3]{\frac{x - 1}{3}} \). Cube roots are crucial in defining inverse functions when original functions involve cubes.

Isolating Variables

Isolating a variable is the process of rearranging an equation so that one variable stands alone on one side of the equation. This is a common necessity in solving equations and finding inverses of functions.

• To isolate a variable generally involves a sequence of algebraic operations like addition, subtraction, multiplication, or division.
• These operations should be done carefully to maintain equality in an equation.

In our example, to isolate \( y \), we first needed to separate \( y^3 \) by subtracting 1 from \( x \), resulting in \( x - 1 = 3y^3 \). We then divided by 3, getting \( y^3 = \frac{x - 1}{3} \), and finally, we took the cube root to solve completely for \( y \). Isolating variables is central to unraveling both direct and inverse functional relationships in math.