Problem 22 - Precalculus Essentials

Question

Determine whether each equation defines y as a function of \(x .\) $$x+y^{3}=27$$

Step-by-Step Solution

Answer

Yes, the equation \(x+y^{3} = 27\) defines \(y\) as a function of \(x\), because for every \(x\) there is only one corresponding \(y\) value.

Step 1: Rewrite the equation

Start by recognizing the goal and rearranging the equation in the terms of \(y\). This can be done by subtracting \(x\) from both sides. So, the equation \(x+y^{3}=27\) can be rewritten as \(y^{3} = 27-x\).

Step 2: Applying the cube root

After getting \(y^{3} = 27-x\), it's necessary to apply the cube root to both sides of the equation in order to isolate \(y\). Thus, the equation becomes \(y = \sqrt[3]{27-x}\).

Step 3: Checking for function

By observing the simplified equation, see that for every \(x\) input, only one \(y\) output would result. Therefore, the equation is indeed a function.

Key Concepts

Function of x

When we talk about a 'function of x', we're discussing a special relationship between two variables, typically represented as 'x' and 'y'. In this context, 'x' is the independent variable, and 'y' is the dependent variable. The idea is that for every value of 'x', there is exactly one corresponding value of 'y'. This one-to-one relationship underlines what it means to be a function. For example, in an equation like the one presented, determining if 'y' is a function of 'x' involves checking if each 'x' value has a single 'y' output.

Understanding this concept is fundamental to mathematics because functions are used to describe real-world phenomena, calculate values, and establish connections between variable quantities.

Cube Root

The cube root is a critical mathematical operation used to find a number that, when multiplied by itself three times, gives the original number. Symbolically, it is represented as \( \sqrt[3]{a} \) where \(a\) is the number you want to find the cube root of. In the given exercise, after the equation has been rearranged to \(y^{3} = 27 - x\), it is necessary to find a number that 'cubed' will give us \(27-x\). Applying the cube root to both sides of the equation is how we isolate 'y', reversing the cubing process.

Isolating y

Isolating 'y' in an equation is a fundamental algebraic skill. It involves manipulating the equation so 'y' comes out alone on one side of the equals sign. This allows you to solve for 'y' and understand how it changes with 'x'. In practical terms, isolating 'y' often means performing operations such as addition, subtraction, multiplication, division, or, in our case, taking the cube root on both sides of the equation. The goal is to leave 'y' by itself without any coefficients or other terms attached to it.

In our exercise, to isolate 'y', we rewrite \(y^{3} = 27 - x\) to \(y = \sqrt[3]{27 - x}\) by taking the cube root. This gives us a clear picture of 'y' in terms of 'x' only.

Equation Rearrangement

Equation rearrangement involves the process of moving terms and operations around in an equation to achieve a desired form, often to isolate a specific variable. It is a fundamental skill in algebra, allowing us to solve equations and understand how variables depend on each other. In our exercise, the original equation \(x + y^{3} = 27\) was rearranged by subtracting 'x' from both sides to isolate the \(y^{3}\) term.

This is a common first step in solving for a variable: altering the equation's structure without changing its meaning or the balance between its two sides. Once rearranged suitably, as we have seen, we can apply other operations like the cube root to solve for 'y'.