Problem 22 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find \(f(x)\) for the indicated values of \(x\), if possible. (b) Find the domain of \(f\). $$ f(x)=2 x-1 \text { for } x=8,-1 $$

Step-by-Step Solution

Answer

(a) \(f(8)=15\), \(f(-1)=-3\). (b) Domain: all real numbers.

Step 1: Substitute and Solve for x = 8

The function is given as \( f(x) = 2x - 1 \). To find \( f(8) \), substitute \( x = 8 \) into the equation:\[f(8) = 2(8) - 1 = 16 - 1 = 15\]

Step 2: Substitute and Solve for x = -1

Substitute \( x = -1 \) into the function \( f(x) = 2x - 1 \):\[f(-1) = 2(-1) - 1 = -2 - 1 = -3\]

Step 3: Determine the Domain of f(x)

The function \( f(x) = 2x - 1 \) is a linear function. Linear functions are defined for all real values of \( x \). Thus, the domain of \( f(x) \) is the set of all real numbers.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial in determining where the function exists or is defined. For any given function, its domain includes all the possible input values (commonly represented by \( x \)) for which the function can output a real number. In simpler terms, it tells us which numbers we can plug into the function without breaking any mathematical rules.

For linear functions like \( f(x) = 2x - 1 \), the domain is quite straightforward. Linear functions are defined by a straight line, and they do not have restrictions like fractions or square roots that could limit possible \( x \) values. Thus, the domain of a linear function is the entire set of real numbers. This means you can substitute any real number for \( x \) into the function, and you'll always get a real number as an output.

In summary, for the function \( f(x) = 2x - 1 \), the domain is all real numbers, often denoted as \( \mathbb{R} \). This tells you that you have the freedom to choose any number to substitute into \( x \).

Real Numbers

Real numbers play a vital role in the world of functions, and understanding them is fundamental to grasping function concepts like domain and range. Real numbers include every possible number you might encounter in the real world, encompassing whole numbers, fractions, and irrational numbers such as \( \sqrt{2} \) or \( \pi \).

Mathematically, the set of real numbers is denoted by \( \mathbb{R} \). This set includes both positive and negative numbers, along with zero, ensuring a continuous spectrum of values without any gaps. This continuity makes real numbers perfect candidates for describing one-dimensional quantities in geometry or algebra.

• Whole Numbers: Include zero, positives (1, 2, 3,…), and negatives (-1, -2, -3,…).
• Fractions: Numbers that represent parts of a whole, like \( \frac{1}{2} \) or \( \frac{3}{4} \).
• Irrational Numbers: Such as \( \pi \) or \( \sqrt{2} \), which cannot be expressed as simple fractions.

When dealing with linear functions like \( f(x) = 2x - 1 \), all these numbers can be used as inputs because the domain covers all real numbers.

Substituting Values in Functions

Substituting values into a function is a straightforward process that allows you to find specific outputs for given inputs. The process involves replacing the \( x \)-variable in the function's equation with a chosen number and then simplifying the expression.

Let's look at the provided example, using the function \( f(x) = 2x - 1 \). Suppose you want to find \( f(8) \). Here's how it's done:

• Start by substituting \( 8 \) for \( x \) in the function: \( f(8) = 2 \times 8 - 1 \).
• Simplify the expression: \( 2 \times 8 = 16 \). So, \( f(8) = 16 - 1 = 15 \).

Similarly, if you substitute \( x = -1 \):

• Replace \( x \) with \( -1 \): \( f(-1) = 2 \times (-1) - 1 \).
• Simplify it: \( 2 \times (-1) = -2 \), hence \( f(-1) = -2 - 1 = -3 \).

By substituting different values, you can see how the function behaves and understand its output for various inputs. This substitution is a key part of working with and analyzing functions.