Problem 49 - College Algebra Essentials

Question

(Hint for Exercises \(49-52:\) To solve for a variable involving an nth root, raise both sides of the equation to the nith power: \(\left.(\sqrt[n]{y})^{n}=y .\right)\) $$f(x)=\sqrt{x-1}$$

Step-by-Step Solution

Answer

The function \(f(x) = \sqrt{x - 1}\) is zero when \(x = 1.\)

Step 1: Write down the function

The function is \(f(x) = \sqrt{x-1}\). The task is finding when the function equals to zero.

Step 2: Setup the equation

Set the function equal to zero to solve for \(x\). This results in the equation \(0 = \sqrt{x-1}\).

Step 3: Square both sides of the equation

In order to eliminate the square root, square both sides of the equation: \(0^2 = (\sqrt{x-1})^2\). This simplifies to \(0 = x - 1\).

Step 4: Solve for \(x\)

After simplification, solve for \(x\). In this case, add 1 to both sides of the equation to isolate \(x\). The solution is \(x = 1.\)

Key Concepts

Understanding the nth Root

The nth root is a fundamental concept in algebra. It represents a number which, when multiplied by itself a certain number of times (n times), results in a given number. For example, the cube root of 8 is 2 because multiplying 2 by itself three times gives 8 (i.e., \(2 \times 2 \times 2 = 8\)). Similarly, an nth root of a number \(y\) is expressed as \(\sqrt[n]{y}\).

In solving algebraic equations, understanding and using the nth root is crucial. It allows us to simplify expressions and ultimately find solutions to equations involving roots. If you have an equation \(\sqrt[n]{y} = a\), you can find \(y\) by raising both sides of the equation to the power of \(n\), resulting in \(y = a^n\). This method was applied in the original exercise to simplify \(\sqrt{x-1}\).

Squaring Equations to Eliminate Roots

Squaring an equation is a useful algebraic technique when you want to eliminate square roots. When we square both sides of an equation that includes a square root, we simplify the problem significantly. In mathematical terms, if you have \(\sqrt{x-1} = 0\), you can square both sides to eliminate the square root, resulting in \((\sqrt{x-1})^2 = 0^2\), which simplifies to \(x-1 = 0\).

Eliminating square roots through squaring is vital as it turns a more complex problem into a simple linear equation. However, it's essential to check possible extraneous solutions because squaring can introduce solutions that do not satisfy the original equation. Always verify the solution by substituting back into the original equation.

Solving Equations to Find Solutions

Solving equations involves finding the values of variables that make the equation true. The solution process often requires a series of operations like addition, subtraction, multiplication, division, or other algebraic manipulations.

• Begin by simplifying the equation if necessary.
• Perform operations to isolate the variable of interest. This can include adding or subtracting terms from both sides.
• If the variable is within a square root or another function, applying inverse operations (like squaring or finding roots) may be necessary.

For instance, with the equation \(x-1=0\), we added 1 to both sides to isolate \(x\), giving us \(x=1\). This step-by-step approach ensures that the solution satisfies the criteria of the original equation.

Functions and Their Role in Equations

Functions play a central role in algebra by describing the relationship between variables. A function like \(f(x) = \sqrt{x-1}\) defines an operation on the input \(x\) to produce an output. Functions can be used to model real-world scenarios or theoretical problems and are foundational elements in many areas of mathematics.

When solving functions set to equal a specific value (like zero), you effectively find the input values that produce that specific output through the function's prescribed operations. For \(f(x) = \sqrt{x-1}\), setting it to zero means finding the \(x\) value satisfying the equation \(\sqrt{x-1} = 0\). This leads us through processes like squaring to solve the equation and find that \(x = 1\). Such practices highlight the importance and utility of understanding and manipulating functions in algebra.