Question

\(f(x)=\sqrt{2 x^2+x+1}\)

Short answer

The function \(f(x) = \sqrt{2 x^2+x+1}\) helps us understand the relationship between 'x' and the output of the function. For example, when \(x = 1\), \(f(x) = 2\), and when \(x = 2\), \(f(x) = \sqrt{11}\).

Step-by-step solution

Understand the Function

The given function is \(f(x)=\sqrt{2 x^2+x+1}\). This means that for every input value of x, the function will output the square root of \(2x^2 + x + 1\).

Choose a Value for x

Let's choose an arbitrary value for x to understand how the function works. For simplicity, let's take \(x = 1\).

Substitute and Calculate

Substitute \(x = 1\) into the function: \(f(1)=\sqrt{2(1)^2+1+1} = \sqrt{2 + 1 + 1} = \sqrt{4} = 2\). So when \(x = 1\), the output of the function \(f(x)\) will be 2.

Try Another Value for x

Let's input another value. For example, \(x = 2\). Substitute \(x = 2\) into the function: \(f(2)=\sqrt{2(2)^2+2+1} = \sqrt{2*4 + 2 + 1} = \sqrt{8+2+1} = \sqrt{11}\). So when \(x = 2\), the function \(f(x)\) is equal to \(\sqrt{11}\).

Key concepts

Understanding Functions

Functions are fundamental in algebra and serve as a bridge that connects every input with a corresponding output. In the context of algebra 2, a radical function like f(x)=\(\sqrt{2 x^2+x+1}\) is one where the output involves taking the square root of a polynomial expression.

Think of a function as a machine: you feed it an input value, and it processes this input according to a specific rule—in this case, the rule involves a radical operation, which means computing the square root. Understanding a radical function requires recognizing the components involved: the radical sign (\(\sqrt{ }\)), the expression underneath this sign known as the radicand (\(2x^2 + x + 1\)), and the variable (\(x\)) for which the function is solved. This understanding is crucial when we begin to modify the function by substituting values into it.

Substituting Values in Functions

Substituting values into functions is akin to testing how the function behaves with different inputs. It's essential for understanding the range and nature of the function's outputs. To substitute a value, simply replace the variable (\(x\)) in the function with the value you are interested in.

For instance, when we substitute \(x = 1\) into the function f(x)=\sqrt{2 x^2+x+1}, we replace every appearance of \(x\) with 1 and then follow the order of operations—first squaring the \(x\), then multiplying by 2, adding the next sequential terms, and lastly taking the square root of the entire expression.

Substitution Example:

• Step 1: Replace \(x\) with 1: \(2(1)^2 + 1 + 1\)
• Step 2: Perform multiplication and addition: \(2 + 1 + 1\)
• Step 3: Compute the square root: \(\sqrt{4}\), which simplifies to 2.

By performing similar substitutions with different values, you can obtain an array of outputs, allowing you to better grasp the behavior and graph of the function.

Radicals and Square Roots

In algebra, a radical expression involves roots, with the square root being one of the most common types. The square root, symbolized by \(\sqrt{ }\), is a function itself that asks for which number, when multiplied by itself, gives us the radicand.

Understanding radicals is not just about computation—it's also about recognizing the type of solutions you might get. For instance, the square root of a positive number will yield both a positive and negative result because both options, when squared, return the original number. This is known as the principle square root. However, in function notation, the principal, or positive, root is typically considered.

Key Points About Radicals:

• The radicand must be positive or zero for real square roots.
• Radical functions involve variables within the radicand, which makes simplifying expressions more involved.
• When dealing with equations, consider both positive and negative square roots, but in function contexts, stick with the principal root unless stated otherwise.

Remembering these points is helpful when you encounter complex radical functions in algebra 2. Each time you see a radical, assess the radicand carefully, as the nature of this inner expression determines the types of numbers you will encounter as outputs from the square root operation.