Problem 34 - Algebra and Trigonometry

Question

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=6 x-18 ; \quad f\left(\frac{x}{3}\right), \frac{f(x)}{3} $$

Step-by-Step Solution

Answer

\( f\left(\frac{x}{3}\right) = 2x - 18 \) and \( \frac{f(x)}{3} = 2x - 6 \).

Step 1: Evaluate f(x/3)

To find the expression for \( f\left(\frac{x}{3}\right) \), substitute \( \frac{x}{3} \) into the function \( f(x) = 6x - 18 \). This gives: \[f\left(\frac{x}{3}\right) = 6\left(\frac{x}{3}\right) - 18 = 2x - 18.\]Simplify to get: \[f\left(\frac{x}{3}\right) = 2x - 18.\]

Step 2: Evaluate f(x)/3

To find \( \frac{f(x)}{3} \), take the function \( f(x) = 6x - 18 \) and divide every term by 3:\[\frac{f(x)}{3} = \frac{6x - 18}{3} = 2x - 6.\]Simplify to get:\[\frac{f(x)}{3} = 2x - 6.\]

Key Concepts

Algebraic Simplification

Algebraic simplification is a critical skill in mathematics. It involves reducing an expression to its simplest form. This can include combining like terms, eliminating unnecessary components, and ensuring the expression is as straightforward as possible.
Simplification is used to make the expression easier to evaluate or use in further calculations. For example, in the given function\[f\left(\frac{x}{3}\right) = 6\left(\frac{x}{3}\right) - 18\]begins with the insertion of \(\frac{x}{3}\) into the function, which provides the intermediate result of \(2x - 18\). Simplifying algebraic expressions like this helps in recognizing patterns and can often make complex problems easier to solve.

Substitution

Substitution in mathematics refers to the process of replacing a variable in an expression with another value or expression. It's particularly useful for evaluating functions and simplifying expressions.
In our exercise, we substitute \(\frac{x}{3}\) into the original function \(f(x) = 6x - 18\). Substitution allows us to transform \(f(x)\) into \[f\left(\frac{x}{3}\right) = 2x - 18\], making it possible to evaluate and simplify parts of the function.
This technique is essential in calculus and algebra for handling complex equations and expressions, by making them more manageable for calculation and analysis.

Polynomial Functions

Polynomial functions are expressions involving variables raised to positive integer powers summed or subtracted. They play a vital role in algebra as they are foundational to many mathematical concepts.
The function \(f(x) = 6x - 18\) is a linear polynomial function, characterized by terms with degrees of zero or one, making them simple to work with. The simplicity aids in focusing on substitution and simplification to understand function behavior.

• Linear polynomials are the building blocks for more complex polynomial functions.
• Understanding linear polynomial functions lays the groundwork for tackling quadratic, cubic, and higher-degree polynomials.

Linear polynomial functions like the one in our exercise illustrate how polynomial functions can be effectively analyzed and simplified, providing a clearer view of their graphical representation and real-world applications.