Question

Find (a) The domain. (b) The range. $$ y=5 x-1 $$

Short answer

Answer: The domain of the function \(y = 5x - 1\) is all real numbers, represented as \((-\infty, \infty)\). The range of the function is also all real numbers, represented as \((-\infty, \infty)\).

Step-by-step solution

(Step 1: Identify the function type)

The given function is a linear function with the equation \(y = 5x - 1\).

(Step 2: Determine the domain)

The domain of a linear function is all real numbers, which can be represented as \((-\infty, \infty)\). So, the domain of our function is: $$ \text{Domain} = (-\infty, \infty). $$

(Step 3: Analyze the function for range)

The linear function is continuous, and its slope is positive. As \(x\) increases, \(y\) also increases with no end value on either end. So, the outputs of the function can also take on any real number.

(Step 4: Determine the range)

Based on the analysis in Step 3, the range of the function is all real numbers, just like the domain. We can represent the range as: $$ \text{Range} = (-\infty, \infty). $$ In conclusion, we have: (a) The domain of the function \(y = 5x - 1\): $$ \text{Domain} = (-\infty, \infty). $$ (b) The range of the function \(y = 5x - 1\): $$ \text{Range} = (-\infty, \infty). $$

Key concepts

Domain of a Function

The domain of a function is essentially the set of all possible input values (often represented by the variable \(x\)) that will output a valid result for the function. For linear functions like \(y = 5x - 1\), there are no restrictions imposed by the function itself on the input value \(x\). This means you can plug in any real number from negative infinity to positive infinity into the equation and still get a real number as a result. Therefore, the domain of this function is all real numbers. Mathematically, this is represented as:

• Domain: \(( -\infty, \infty )\)

The idea is quite simple: since no value of \(x\) can make the linear equation undefined, every real number is allowed as an input. Hence, linear functions possess an unrestricted domain.
Another way to think about this is that a straight line extends infinitely in both directions on the \(x\)-axis, covering all real numbers.

Range of a Function

Understanding the range of a function involves identifying all possible output values, usually represented by the variable \(y\). For linear functions, the range is also very straightforward to determine. As you input every real number into the function \(y = 5x - 1\), the outputs are similarly unrestricted, spanning all real numbers. This happens because a linear function graph is a straight line that never ends or is interrupted. Hence, it can achieve any real number for \(y\) as well.

• Range: \(( -\infty, \infty )\)

It's worth noting that the slope of the line, which is 5 in this case, indicates how \(y\) changes with \(x\). But regardless of how steep or flat the line is, because it extends indefinitely, every real number can eventually be a \(y\)-value. This consistent behavior is what makes linear functions special, as they have the same domain and range: all real numbers.

Real Numbers

Real numbers are a fundamental concept in mathematics, encompassing almost every number you encounter in daily life. They include:

• Positive and negative integers (e.g., -3, 0, 7)
• Fractions and decimals (e.g., 1/2, 0.75)
• Irrational numbers that can't be expressed as simple fractions (e.g., \(\pi\), \(\sqrt{2}\))

When we say a function has a domain or range of all real numbers, we mean that the function's inputs and outputs can be any of these types of numbers. Real numbers are represented mathematically in interval notation as \(( -\infty, \infty )\), indicating that there are no bounds to the smallest or largest numbers.This broad definition is essential for understanding the behavior of functions like linear ones. Since linear functions deal with unrestricted real number inputs and outputs, they maintain simple yet powerful mathematical models applicable in various real-world scenarios.