Problem 28 - 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

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

Step 1: Understanding the Function

The given function is \( f(x) = 6x - 18 \). This function is linear, with a slope of 6 and a y-intercept of -18.

Step 2: Evaluate \( f\left(\frac{x}{3}\right) \)

To evaluate \( f\left(\frac{x}{3}\right) \), substitute \( \frac{x}{3} \) into the function. We start by replacing every \( x \) in \( f(x) \) with \( \frac{x}{3} \): \[ f\left(\frac{x}{3}\right) = 6\left(\frac{x}{3}\right) - 18 \] Simplify: \[ f\left(\frac{x}{3}\right) = 2x - 18 \]

Step 3: Evaluate \( \frac{f(x)}{3} \)

Now, evaluate \( \frac{f(x)}{3} \) by dividing the entire function \( f(x) \) by 3: \[ \frac{f(x)}{3} = \frac{6x - 18}{3} \] Simplify each term: \[ \frac{6x}{3} - \frac{18}{3} = 2x - 6 \]

Step 4: Simplified Expressions

The simplified expressions are: \( f\left(\frac{x}{3}\right) = 2x - 18 \) \( \frac{f(x)}{3} = 2x - 6 \) Both expressions have been computed correctly. The first expression shows how \( f \) changes when its argument is scaled, while the second shows the function when divided entirely by 3.

Key Concepts

Linear Function

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. Linear functions have the general form \( f(x) = mx + b \).

• \( m \) is the slope: It indicates how steep the line is and the direction it goes (up or down).
• \( b \) is the y-intercept: It is the point where the line crosses the y-axis.

In our exercise, the function \( f(x) = 6x - 18 \) is linear because it can be expressed in this form. Here, the slope (\( m \)) is 6, meaning the line increases steeply, and the y-intercept (\( b \)) is -18, indicating the line crosses the y-axis at (0, -18).
Understanding these components will help us grasp how changing elements of the function affects its graph.

Substitution in Functions

Substitution in functions involves replacing the input variable \( x \) with another expression or value to determine the output. This process can help us observe how a function behaves under different conditions.
In our example, we are asked to find \( f\left(\frac{x}{3}\right) \) by substituting \( x \) with \( \frac{x}{3} \) into the original function \( f(x) = 6x - 18 \).

• Substitute: Replace \( x \) in the function with \( \frac{x}{3} \).
• Evaluate: The expression then becomes \( f\left(\frac{x}{3}\right) = 6\left(\frac{x}{3}\right) - 18 \).
• Simplify: Using arithmetic operations, simplify the expression to \( 2x - 18 \).

This process shows how the output changes when the input is scaled. Using substitution, we can evaluate functions with various inputs to see different outcomes.

Simplifying Expressions

Simplifying expressions involves making them more concise while retaining their value. It often includes combining like terms or reducing fractions.
In our exercise, we first evaluate the expression \( f\left(\frac{x}{3}\right) \) which simplifies to \( 2x - 18 \). Then, we simplify \( \frac{f(x)}{3} \).

• Start with \( \frac{f(x)}{3} = \frac{6x - 18}{3} \): Dividing each term by 3.
• \( \frac{6x}{3} - \frac{18}{3} = 2x - 6 \): Simplify each term separately.

The simplified expressions, \( f\left(\frac{x}{3}\right) = 2x - 18 \) and \( \frac{f(x)}{3} = 2x - 6 \), reveal more straightforward versions that are equivalent to the original forms. Simplifying not only improves clarity but is essential in solving equations and comparing expressions effectively.