Problem 20 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=3 \sqrt{x} ; 9$$

Step-by-Step Solution

Answer

The function \(y=3 \sqrt{x}\) evaluates to 9 when \(x = 9\).

Step 1: Substitute the given value

The first step is to substitute \(x = 9\) into the given function \(y=3 \sqrt{x}\). So, we have \(y = 3 \sqrt{9} \).

Step 2: Evaluate the square root

The second step involves evaluating the square root. The square root of 9 is 3. So, this gives us \(y = 3 \times 3\).

Step 3: Evaluate the multiplication

The final step is to carry out the multiplication. Therefore, the equation becomes \(y = 9\).

Key Concepts

Square Root

The square root, represented as \( \sqrt{x} \), is a value that, when multiplied by itself, gives the original number \( x \). In other words, if \( a^2 = x \), then \( a \), is the square root of \( x \). For example, since \( 3^2 = 9 \), \( 3 \), is the square root of \( 9 \).

When evaluating square roots, it's essential to remember that there is both a positive and a negative root, often denoted as \( \pm \sqrt{x} \). However, in most basic functions, we only consider the principal (positive) square root. It's also good practice to simplify square roots when possible to make further calculations easier. For instance, knowing that \( \sqrt{9} = 3 \) simplifies evaluating functions that include square roots.

Substitution Method

replace every instance of \( x \) in the equation with that particular value. For instance, if you have a function such as \( y = 3 \sqrt{x} \) and you want to evaluate it at \( x = 9 \), you substitute \( 9 \) for \( x \), resulting in \( y = 3 \sqrt{9} \).

This method not only simplifies the process of function evaluation but also helps in understanding how different values of the input \( x \) affect the output \( y \) of the function. It's essential to perform substitution accurately to avoid any errors in the final result.

Mathematical Functions

Mathematical functions are fundamental constructs in algebra and calculus that assign every input value exactly one output value. They can be visualized as 'machines' where you insert a number \( x \) and get out another number \( y \), based on a specific set of rules defined by the function. For example, the function \( y = 3 \sqrt{x} \) will output a value of \( y \) that is three times the square root of the input \( x \).

Recognizing the different types of functions, including linear, quadratic, polynomial, and radical functions, allows you to apply the appropriate methods for evaluation, like the substitution method for calculating the value of \( y \) when \( x \) is known. A strong grasp of functions is invaluable for solving complex problems and understanding the behavior of various mathematical models.