Problem 8 - Algebra for College Students

Question

If \(f(x)=\sqrt{3 x+2}\), find \(f\left(\frac{14}{3}\right), f(10)\), and \(f\left(-\frac{1}{3}\right)\).

Step-by-Step Solution

Answer

The values are 4, \(4\sqrt{2}\), and 1, respectively.

Step 1: Identifying the Function

We are given the function \( f(x) = \sqrt{3x + 2} \). This means that for any input \( x \), the output is the square root of \( 3x + 2 \).

Step 2: Evaluating \( f \left(\frac{14}{3}\right) \)

To find \( f \left(\frac{14}{3}\right) \), substitute \( \frac{14}{3} \) into the function: \( f \left(\frac{14}{3}\right) = \sqrt{3 \left(\frac{14}{3}\right) + 2} \). Simplifying inside the square root gives \( \sqrt{14 + 2} = \sqrt{16} = 4 \).

Step 3: Evaluating \( f(10) \)

Substitute \( 10 \) into the function: \( f(10) = \sqrt{3 \times 10 + 2} \). Simplifying inside the square root gives \( \sqrt{30 + 2} = \sqrt{32} = 4 \sqrt{2} \).

Step 4: Evaluating \( f \left(-\frac{1}{3}\right) \)

Substitute \( -\frac{1}{3} \) into the function: \( f \left(-\frac{1}{3}\right) = \sqrt{3 \times \left(-\frac{1}{3}\right) + 2} \). Simplifying inside the square root gives \( \sqrt{-1 + 2} = \sqrt{1} = 1 \).

Key Concepts

Evaluating Functions

Evaluating functions is similar to solving a puzzle where each piece fits in a specific spot. In mathematics, when we "evaluate" a function, we are essentially determining what output we get for a specific input.
To evaluate a function like \( f(x) = \sqrt{3x + 2} \), we replace the variable \( x \) with the number we want to evaluate. For example, if we want to find \( f(5) \), we substitute \( 5 \) for \( x \):

• Write: \( f(5) = \sqrt{3 \, \times \, 5 + 2} \)
• Calculate inside the square root: \( f(5) = \sqrt{15 + 2} \)
• Final answer: \( f(5) = \sqrt{17} \)

By following these steps, you can systematically find the output of any function for any given input.

Square Roots

Square roots are the opposite of squaring a number and are vital in understanding functions in algebra. A square root of a number \( a \) is a value that, when multiplied by itself, gives \( a \). The symbol \( \sqrt{} \) indicates a square root, and the most familiar is \( \sqrt{16} \), which equals \( 4 \), because \( 4 \times 4 = 16 \).
Square roots play a significant role in functions that involve a radical (like \( \sqrt{3x + 2} \)). When simplifying:

• Calculate inside the root first.
• Find the square root of the resulting number.

Remember, not all numbers have neat square roots. For example, \( \sqrt{17} \) is irrational, meaning it cannot be expressed as a simple fraction, and it's approximately \( 4.123 \). Understanding square roots is key to solving and simplifying algebraic expressions involving radical signs.

Function Notation

Function notation is a way of writing algebraic equations that clearly shows the input and output relationship. When you see \( f(x) \), this means "the function \( f \) in terms of \( x \)." The letter inside the parenthesis (in this case, \( x \)) represents the input value.
The notation becomes a way to denote that for any input \( x \), there is a specific output called \( f(x) \). This is crucial because:

• It helps to specify which variable is the independent one (input).
• It clarifies which function you are working with, especially when dealing with multiple functions.

To use function notation, you simply replace the \( x \) with the desired input. If evaluating multiple inputs, like in our exercise, substitute each value into \( f(x) \) to determine their unique outputs. Function notation keeps your work organized and ensures consistency across mathematical problems.