Problem 25 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{21-2 x} ;-2$$

Step-by-Step Solution

Answer

The function \(y=\sqrt{21-2 x}\) evaluated at \(x=-2\) is \(5\).

Step 1: Understand the Function

The given function is a square root function, denoted by \(y=\sqrt{21-2 x}\).

Step 2: Substitution

Substitute the given value of \(x = -2\) into the function. The new equation would be \(y=\sqrt{21-2(-2)}\).

Step 3: Simplify the equation

By multiplying \(2\) by \(-2\), the equation simplifies into \(y=\sqrt{21+4}\), which further simplifies to \(y=\sqrt{25}\).

Step 4: Calculate the root

Calculating the square root of \(25\) yields \(5\). Therefore, \(y=5\).

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental concept in algebra, which essentially asks the question: what number squared equals the given value? Visually, it is represented as \(y = \sqrt{x}\), where \(y\) is the square root of \(x\). In our exercise, the square root function is slightly more complex, \(y=\sqrt{21-2x}\). This means for any value of \(x\), we're looking for a number that, when squared, gives us the result of \(21-2x\).

When evaluating such functions, it's crucial to pay close attention to the input inside the square root, known as the radicand. Ensure the final value inside the square root is non-negative, as square roots of negative numbers aren't real numbers (in basic algebra, we consider only real roots). Finally, remember that square root functions graph as a parabola opening upwards or sideways, reflecting their systematic nature in increasing or decreasing.

Applying the Substitution Method

The substitution method in algebra is a technique used to evaluate expressions and functions. It involves replacing a variable in an equation with its corresponding value and simplifying the equation thereafter. In our exercise, we are given \(x = -2\), and we substitute this value into our square root function.

To effectively use this method, follow these steps: Replace the variable \(x\) with the given number, ensuring to keep the order of operations intact. This means, in our case, the negative sign in front of the substituted value must be accounted for in the computation, as it changes the operation from subtraction to addition. Substitution is a powerful algebraic tool because it makes evaluating complex functions possible by simplifying them down to more manageable arithmetic.

Simplifying Expressions Effectively

Simplifying expressions is a crucial process in algebra to reduce equations to their simplest form, which makes evaluating or solving them much easier. Simplifying may involve combining like terms, reducing fractions, or, as seen in our exercise, performing addition or subtraction before squaring.

The steps we take include multiplying variables with numbers, recognizing inverse operations that eliminate each other, and remembering that the square root of a perfect square results in a positive whole number. For example, when we simplified the expression \(\sqrt{21-2(-2)}\) to \(\sqrt{25}\), we combined like terms and recognized the perfect square. Simplification often requires attention to detail, ensuring that every step follows mathematical rules and leads to the simplest possible expression without altering its value.