Problem 47 - Elementary and Intermediate Algebra

Question

Find \(f(3)\) and \(f(-1) .\) See Example 4. $$ f(x)=3 x $$

Step-by-Step Solution

Answer

\(f(3) = 9\) and \(f(-1) = -3\).

Step 1: Identify the Function

The given function is linear, and it is defined as \( f(x) = 3x \). This function indicates that for any input \( x \), the output is found by multiplying \( x \) by 3.

Step 2: Substitute for \( f(3) \)

To find \( f(3) \), substitute 3 for \( x \) in the function: \[ f(3) = 3(3) = 9 \] This means that when the input is 3, the output is 9.

Step 3: Substitute for \( f(-1) \)

To find \( f(-1) \), substitute -1 for \( x \) in the function: \[ f(-1) = 3(-1) = -3 \] This means that when the input is -1, the output is -3.

Key Concepts

Linear Functions

Linear functions are a key concept in algebra and mathematics. They are called "linear" because they graph as straight lines when plotted on a coordinate plane. A linear function is typically expressed in the form:

\[ f(x) = mx + b \]

In this expression,:

• \( m \) represents the slope, which shows the rise over the run or how steep the line is.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

The function given in our problem is \( f(x) = 3x \). This is a linear function with:

• Slope \( m = 3 \)
• Y-intercept \( b = 0 \) because there is no added constant at the end.

This means the line goes through the origin (0,0) and increases by 3 units vertically for every 1 unit it moves horizontally.

Function Notation

Function notation is used to express the output of a function for any given input. It is a way of writing functions that makes major operations easier to understand and manage. For example, in the expression:

\[ f(x) = 3x \]

• The letter \( f \) is just the name of the function.
• The \( x \) in parentheses indicates the variable or the input of the function.

The expression \( f(x) \) represents "the output of the function \( f \) when the input is \( x \)." This notation allows mathematicians to succinctly express how inputs are transformed into outputs, which is extremely useful in various calculations and analyses. By using function notation, it becomes straightforward to plug values into the function to find corresponding outputs.

Substitution Method

The substitution method is a simple yet powerful technique used to evaluate functions. It involves replacing the variable in a function's equation with a specific value. This allows you to find out what the function's output is for that specific input.

To apply the substitution method in this context:

• Take the function \( f(x) = 3x \).
• To find \( f(3) \), substitute 3 in place of \( x \) so it becomes \( 3(3) \), which equals 9.
• To find \( f(-1) \), substitute -1 in place of \( x \) so it becomes \( 3(-1) \), which equals -3.

This method is useful because it provides a clear, mechanical way of determining outputs for different inputs. It is widely applied in algebra to simplify problems and verify solutions.