Problem 34 - Algebra A Combined Function

Question

Find the inverse of each one-to-one function. $$ f(x)=\sqrt[3]{x+1} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( f(x) = \sqrt[3]{x+1} \). This function represents a cube root transformation of \( x+1 \). Our goal is to find its inverse \( f^{-1}(x) \).

Step 2: Replace f(x) with y

To find the inverse function, start by replacing \( f(x) \) with \( y \). So, we have \( y = \sqrt[3]{x+1} \).

Step 3: Swap x and y

To find the inverse, swap the roles of \( x \) and \( y \). So, the equation becomes \( x = \sqrt[3]{y+1} \).

Step 4: Solve for y

Cube both sides of the equation to eliminate the cube root: \( x^3 = y + 1 \).

Step 5: Isolate y

Subtract 1 from both sides to isolate \( y \): \( y = x^3 - 1 \).

Step 6: Write the Inverse Function

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

Key Concepts

Understanding One-to-One Functions

A one-to-one function is a special type of function where each element of the domain is paired with a unique element of the range. That means no two different inputs can lead to the same output. One-to-one functions are important in finding inverses because they ensure that the inverse of the function will also be a function. This is because each output of the function corresponds to exactly one input. For example:

• Consider the function \( f(x) = x + 2 \). Here, each input \( x \) gives a unique output, like \( 1 \rightarrow 3 \) and \( 2 \rightarrow 4 \).
• If a function is not one-to-one, such as \( f(x) = x^2 \), it can happen that two different inputs give the same output, like \( 2 \) and \(-2\) both giving \( 4 \).

To verify a function is one-to-one, you can use the Horizontal Line Test. If no horizontal line intersects the graph of the function more than once, it is one-to-one.Understanding this concept confirms that if you're reversing \( f(x) = \sqrt[3]{x+1} \), there will be a unique way to find the inverse.

Exploring the Cube Root Function

The cube root function is a mathematical operation where you determine a value that, when cubed, equals the original number. In this exercise, we are working with \( f(x) = \sqrt[3]{x+1} \). The cube root is denoted by \( \sqrt[3]{x} \), and indicates the number \( y \) such that \( y^3 = x \).This function is unique because:

• It is available for all real numbers, unlike square roots which are only defined for non-negative numbers.
• The output can be both positive and negative, given the nature of cubing numbers.
• The graph of the cube root function is always increasing, confirming it is one-to-one.

By understanding the cube root function, reversing \( f(x) \) involves transitioning from the transformation \( x + 1 \) inside the root and working backwards through cube powers.

Decoding Function Transformation

Function transformation involves changing a basic function's position or shape on a graph without altering the fundamental relationship between its inputs and outputs. Transformations include translations, dilations, reflections, and rotations. In our context:

• The function \( f(x) = \sqrt[3]{x+1} \) is a transformation of the basic cube root function \( g(x) = \sqrt[3]{x} \).
• The \( +1 \) inside the cube root shifts the graph horizontally to the left by 1 unit. This compensates for the input value in the function.
• Understanding this shift is crucial for finding and interpreting the inverse function, as it guides the process of reversing the function's effect.

Recognizing transformations allows you to predict how modifying \(x\) and \(y\) in equations affects the entire graph, thus simplifying the process of finding an inverse function.