Problem 52 - Precalculus Essentials

Question

To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \(\left.(\sqrt[n]{y})^{n}=y .\right)\). $$f(x)=\sqrt[3]{x-1}$$

Step-by-Step Solution

Answer

To solve for x from the function \(f(x)=\sqrt[3]{x-1}\), we can rewrite the function by raising both sides to the third power. Isolate x by adding 1 to both sides of the equation, rendering it \(x = f^{3}(x)+1\).

Step 1: Rewrite Function

We start by rewriting the function. Our function is \(f(x)=\sqrt[3]{x-1}\). When we rewrite the function in terms of x we get \(f^{3}(x)=(x-1)\).

Step 2: Isolate the Variable

Next, we isolate x by adding 1 to both sides of the equation, yielding: \(f^{3}(x)+1=x\). It is important here to remember that whatever operation is performed on one side of the equation has to be done on the other side to keep it balanced.

Step 3: Solve for x

With x now isolated, we can now solve for it given any value of the function. \(x = f^{3}(x)+1\)

Key Concepts

Solving Equations

When solving equations, our main goal is to find the value(s) of the unknown variable that makes the equation true. The process requires careful manipulation of the equation while maintaining balance on both sides. In the context of the given exercise, balance refers to treating both sides of the equation equally as operations are performed.

If you're trying to solve an equation like \( f(x) = \sqrt[3]{x-1} \), it involves the cube root of an expression. To solve for "\( x \)," steps typically include:

• Isolating the function: Begin by rewriting the function or expression if necessary.
• Raising to a power: Here, you would cube both sides (raise both sides to the third power) because we have defined the cube root.
• Rearranging the equation: Make sure "\( x \)" is on one side of the equation.

This systematic approach helps in isolating the variable \( x \) and eventually solving for it.

Cube Roots

A cube root is a special mathematical operation that involves finding a number which, when multiplied by itself three times, gives back the original number. The cube root of \( x \), denoted \( \sqrt[3]{x} \), reverses the cubing process and holds significance in various algebraic functions.

For example, in the exercise where \( f(x) = \sqrt[3]{x-1} \), the cube root indicates that we're looking for a number whose cube is equal to \( x - 1 \). Understanding cube roots is critical in solving equations involving this operation as it directly pertains to the method of isolating the unknown variable before solving it. By raising the expression to the third power, the cube root and the cubing "cancel out," effectively simplifying the equation to a more manageable form.

Cube roots are handy in various applications such as calculating volumes or when solving polynomial equations.

Functions in Algebra

In algebra, functions are equations that relate an input (often denoted as \( x \)) to an output \( f(x) \). A function shows how each input is paired with exactly one output. For instance, the function \( f(x) = \sqrt[3]{x-1} \) describes how input values lead to output values through the cube root transformation.

Functions are crucial because they allow us to model relationships between quantities in a systematic way. Understanding how to manipulate functions, such as solving them or rewriting them, is essential in algebra. The process involves substitution, isolation of variables, and occasionally, applying specific operations like rooting or exponentiation.

Key aspects of functions include:

• The domain: the set of permissible input values.
• The range: the set of possible output values.
• Inverse functions: functions that "reverse" others.

When dealing with functions in algebra, it's important to carefully analyze their structure and properties to allow for correct manipulation and interpretation.