Problem 14 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(f(x)=6 x-5\) a. \(f(12)\) b. \(f\left(-\frac{1}{2}\right)\) c. \(f(0)\)

Step-by-Step Solution

Answer

The evaluated values are: \(f(12)= 67\), \(f(-1/2)= -8\), \(f(0)= -5\).

Step 1: Evaluate \(f(12)\)

Substitute \(x\) in \(f(x)=6x-5\) with 12. So, \(f(12)= 6 \cdot 12 - 5\). Calculate the multiplication and subtraction to find the value.

Step 2: Evaluate \(f(-1/2)\)

Next, substitute \(x\) in \(f(x)=6x-5\) with -1/2. So, \(f(-1/2)= 6 \cdot (-1/2) - 5\). Again, calculate this to find the resulting value.

Step 3: Evaluate \(f(0)\)

Finally, substitute \(x\) in \(f(x)=6x-5\) with 0. So, \(f(0)= 6 \cdot 0 - 5\). Perform the multiplication and subtraction to find the final result.

Key Concepts

Linear Functions

Linear functions are one of the fundamental concepts in algebra that form a straight line when graphed. They are generally expressed in the form \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Linear functions are called 'linear' because their graph is a line.

In a linear function:

• The slope \(m\) determines how steep the line is. A positive slope means the line goes upward as you move from left to right, while a negative slope means the line goes downward.
• The y-intercept \(b\) is where the line crosses the y-axis. This is the value of the function when \(x=0\).

Understanding linear functions helps in solving many algebraic problems, as these functions have a constant rate of change. This constant change is what makes the graph a straight line, illustrating the predictable nature of linear relationships.

Function Evaluation

Function evaluation involves finding the output of a function for specific input values. It is essentially the process of replacing the variable in the function with a given number, and calculating the result.

For example, to evaluate the function \(f(x) = 6x - 5\), you would:

• Identify the function's formula: \(f(x) = 6x - 5\).
• Choose a value to substitute for \(x\), such as \(x=12\).
• Replace \(x\) with this number to calculate the result: \(f(12) = 6 \cdot 12 - 5\).

This process helps us to understand how the function behaves with different inputs, showing its effect on the output, or the dependent variable. Function evaluation is a practical skill for analyzing mathematical relationships and predicting outcomes based on existing data.

Substitution Method

The substitution method is a straightforward technique used to solve equations by replacing variables with given values. In the context of evaluating functions, substitution involves inserting specific numbers into the function's formula in place of the variable.

To use the substitution method effectively:

• Identify the variable in the function equation, such as \(x\) in \(f(x) = 6x - 5\).
• Insert the given value of the variable into the function. For example, substituting \(x = -\frac{1}{2}\) results in \(f(-\frac{1}{2}) = 6 \cdot (-\frac{1}{2}) - 5\).
• Perform the arithmetic operations following substitution to find the function's value.

This method is crucial as it allows the solving of equations that include variables, helping to find specific values and solutions required in solving algebraic problems. It is essential for working through scenarios where mathematical expressions need precise evaluation based on given inputs.