Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

If \(f(x)=\sqrt{x-4}-3 x\), find \(f(4), f(8)\), and \(f(13)\)

Step-by-Step Solution

Answer

\(f(4) = -12\), \(f(8) = -22\), \(f(13) = -36\).

Step 1: Find f(4)

To find the value of the function when \(x = 4\), we substitute \(x = 4\) into the function \(f(x) = \sqrt{x-4} - 3x\). This gives us: \[ f(4) = \sqrt{4 - 4} - 3 \times 4 = \sqrt{0} - 12 = 0 - 12 = -12 \] Therefore, \(f(4) = -12\).

Step 2: Find f(8)

To calculate \(f(8)\), substitute \(x = 8\) into the function \(f(x) = \sqrt{x-4} - 3x\). This gives us: \[ f(8) = \sqrt{8 - 4} - 3 \times 8 = \sqrt{4} - 24 = 2 - 24 = -22 \] Therefore, \(f(8) = -22\).

Step 3: Find f(13)

To find \(f(13)\), substitute \(x = 13\) into the function \(f(x) = \sqrt{x-4} - 3x\). This gives us: \[ f(13) = \sqrt{13 - 4} - 3 \times 13 = \sqrt{9} - 39 = 3 - 39 = -36 \] Therefore, \(f(13) = -36\).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and numbers where letters are used to represent numbers or quantities. In this exercise, we focus on the function evaluation in algebra. A function like \(f(x)=\sqrt{x-4}-3x\) helps us determine specific values by substituting \(x\) with given numbers. Functions are core in algebraic expressions to model real-world situations. To evaluate \(f(x)\) at different values, replace \(x\) with the desired number and solve the expression.
Key algebra concepts in this particular function involve:

• Using operations like subtraction and multiplication
• Applying rules of order of operations to solve the expression

Practice in algebra involves calculating function values for different \(x\). This process helps strengthen skills in manipulating algebraic symbols and understanding how they interact with numbers in expressions. By switching each \(x\) with values such as 4, 8, and 13, students learn how functions behave with different inputs.

Square Root

Square roots are another significant concept employed here. A square root of a number \(a\) is a value that, when multiplied by itself, gives \(a\). In symbols, if \(b^2 = a\), then \(b\) is the square root of \(a\). Seen throughout the function as \(\sqrt{x-4}\), square roots are typically denoted with a radical symbol \(\sqrt{}\).
Key points about square roots include:

• The square root of 0 is always 0
• Positive numbers have two square roots: positive and negative, but the positive root is often considered the principal square root
• Square root of a number \(x\) is commonly expressed in its simplest form

In the original solution, evaluating \(\sqrt{0}\), \(\sqrt{4}\), and \(\sqrt{9}\) involves determining these specific values, which helps define the behavior of \(f(x)\) at points where changes occur due to the operation.

Substitution Method

The substitution method involves replacing a variable with a number or another expression to simplify or solve algebraic problems. It's a widely used method in algebra to evaluate functions or solve equations. This method is crucial in this exercise.
Steps in substitution applied here are:

• Select the value of \(x\) to substitute in the function, such as 4, 8, or 13
• Replace \(x\) in the function \(f(x) = \sqrt{x-4} - 3x\) with the selected value
• Solve the expression to find the result

Using substitution in various contexts helps understand how variables affect function outcomes. It builds fundamental skills needed not only for basic tests but also for more complex algebraic manipulations in advanced mathematics. Through substitution, students can visualize how changes to \(x\) impact the overall function, moving from unknown to known values.