Problem 22 - College Algebra

Question

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$f(x)=3 x+7$$ a. \(f(4)\) b. \(f(x+1) \quad\) c. \(f(-x)\)

Step-by-Step Solution

Answer

The solved values for \(f(4)\), \(f(x+1)\), and \(f(-x)\) are 19, \(3x+10\), and \(-3x+7\), respectively.

Step 1: Evaluate f(4)

Substitute 4 for x in the function. It becomes \(f(4) = 3*4 + 7\). After carrying out the multiplication, our expression is simplified to \(f(4) = 12 + 7\). Hence, \(f(4) = 19\).

Step 2: Evaluate f(x+1)

Substitute \(x+1\) for x in the function. It becomes \(f(x+1) = 3*(x+1) + 7\). Simplify this to get \(f(x+1) = 3x + 3 + 7\), which simplifies further to \(f(x+1) = 3x+10\).

Step 3: Evaluate f(-x)

Substitute \(-x\) for x in the function. It becomes \(f(-x) = 3*(-x) + 7\). By simplifying, the expression becomes \(f(-x) = -3x + 7\).

Key Concepts

Independent Variable

When dealing with functions, understanding the independent variable is crucial. In simple terms, the independent variable is the input of the function, usually represented by the letter "x." In our exercise, the independent variable changes based on what values we're evaluating: it could be a specific number, like 4; an expression, such as \(x+1\); or even a negative value \(-x\).

• The independent variable is what we have control over; we choose its value.
• In a function like \(f(x) = 3x + 7\), the 'x' is independent. Changing 'x' affects the output of the function.

Understanding how the independent variable influences the function is essential for evaluating and simplifying expressions.

Function Substitution

Function substitution involves replacing the independent variable with a given value or expression. It's like swapping 'x' with something else. For our example function \(f(x) = 3x + 7\), to find \(f(4)\), substitute 'x' with 4, giving us \(3*4 + 7\).

• Start by locating 'x' in your function.
• Then, replace every 'x' in the function with the new value or expression.
• After substitution, simplify the resulting expression.

Substitution is a powerful tool as it allows us to explore how functions behave for different inputs.

Simplification

In mathematics, simplification is the process of making an expression easier to work with. After substituting values into a function, it's crucial to simplify the expression to find its simplest form. Consider \(f(4)\) from the exercise, which after substitution became \(3*4 + 7\).

• Simplify by performing the multiplication: \(12 + 7\).
• Add the numbers to find the final result: \(19\).

The goal of simplification is to express the function in its simplest form to easily find the outcome or explore function properties.

Linear Functions

Linear functions are one of the most basic types of functions you will encounter. They are characterized by a constant rate of change and can be written in the form \(f(x) = mx + b\), where 'm' is the slope and 'b' is the y-intercept. The function given in the exercise, \(f(x) = 3x + 7\), is a linear function.

• The slope (3 in this case) tells us how steep the line is.
• The y-intercept (7 here) indicates where the line crosses the y-axis.

Understanding linear functions is essential because they represent straightforward relationships where one quantity varies linearly with another.