Question

\(f(x)=\sqrt[3]{x}, g(x)=\sqrt[3]{x+4}-5\)

Short answer

The functions intersect at the point (-121,0).

Step-by-step solution

Set Functions Equal

First, set the functions equal to each other to find where they intersect: \(\sqrt[3]{x} = \sqrt[3]{x+4}-5\).

Simplify the Equation

Next, to simplify the equation, cube both sides to remove the cube root. This results in the equation: \(x = (x+4)-5^3\).

Solve for x

Then we rearrange the equation to solve for x: \(x = (x+4)-125\).

Simplify further to find x

To find the solution for x in the equation \(x = (x+4)-125\), we simplify, resulting in \(x = -121\) or \(x = 29\). Either of these are potential solutions because both values for x will make the equation true.

Checking the solutions

Check the solutions by substituting them back into the equations to ensure both sides of the equation are equal. It is found that only \(x = -121\) satisfies both equations. Therefore, the point of intersection of the functions is at (-121,0).

Key concepts

Cube Roots in Algebra

Understanding cube roots in algebra is crucial when solving equations involving three-dimensional shapes or when dealing with real-life volume problems. In the context of our exercise, both functions,

\(f(x) = \sqrt[3]{x}\) and \(g(x) = \sqrt[3]{x+4}-5\), incorporate cube roots, which are the inverse of cubing a number. To solve a cube root equation like \(\sqrt[3]{x} = a\), we cube both sides to get \(x = a^3\), thus eliminating the cube root.

Importance of Proper Cubing

To correctly solve for \(x\), it's not enough to simply square the number you get after removing the cube root; you must cube it. This is because cubing is the inverse operation of finding a cube root, thereby making it possible to isolate \(x\). In our exercise, when cubing both sides, it is crucial to recognize that \(5^3\) or \(125\) comes from the cubed number on the right side of the equation, not from \(5\) added after the cube root operation. This is a critical step that can often lead to mistakes if overlooked.

Equations of Functions

Equations of functions represent relationships between variables. Functions like \(f(x)\) and \(g(x)\) might appear complex with cube roots, but fundamentally they follow the principles of any ordinary function. The solutions to these functions are the 'x' values that satisfy both equations simultaneously, or in other words, where these functions intersect on the graph.

Setting Functions Equal

In our step-by-step solution, the key operation was to set \(f(x)\) equal to \(g(x)\), which is a standard procedure to find where two functions intersect. This allows us to equate the two expressions, and following algebraic manipulations, to find the common 'x' values. The elegance of algebra lies in the ability to manipulate these expressions to uncover the unknown values that satisfy both functions, clearly proving their point of intersection.

Algebraic Problem Solving

Algebraic problem solving involves a series of steps that simplify and solve equations. Our exercise provides a clear example of this systematic approach. Beginning with setting functions equal to each other and ending with verifying the solutions, each step is crucial to finding the correct answer.

Verification of Solutions

An essential part of algebraic problem solving, exhibited in the last step of our exercise, is verifying our solutions. After algebraic manipulation, two potential solutions were found: \(x = -121\) and \(x = 29\). By substituting these values back into the original functions, we can confirm that only \(x = -121\) makes both sides of the equation equal. This step ensures that we have not only solved the equation but also validated that the solution is correct within the context of the functions in question. This verification guards against the introduction of extraneous solutions that may result from the algebraic process.