Problem 64 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$h(x)=-8 x-2$$

Step-by-Step Solution

Answer

The domain of the function \(h(x) = -8x - 2\) is all real numbers, represented in interval notation as \((-\infty, \infty)\).

Step 1: Identify Restrictions on x

Since the function is a linear function without any fractions or square roots, there are no restrictions on the values of \(x\).

Step 2: Determine the Domain

Considering that there are no restrictions on the values of \(x\), the domain will include all real numbers. In interval notation, the domain can be expressed as \((-\infty, \infty)\). So, the domain of the function \(h(x) = -8x - 2\) is \((-\infty, \infty)\).

Key Concepts

Linear Functions

Linear functions are among the simplest types of mathematical functions. They can be represented by a straight line on a Cartesian coordinate system.
The standard form of a linear function is given by:

• \( f(x) = ax + b \)

where \(a\) and \(b\) are constants, and \(x\) is the variable. In the exercise example, the function is \( h(x) = -8x - 2 \). Here, \(-8\) is the slope, which tells us how steep the line is, and \(-2\) is the y-intercept, which tells us where the line crosses the y-axis.
Linear functions can extend infinitely in both directions without any breaks or limits in the domain. This makes them ideal for understanding basic concepts in algebra and calculus.
They do not involve fractions or square roots of \(x\), thus they do not impose any restrictions on the domain.

Real Numbers

The term 'real numbers' refers to the set of values that includes both rational and irrational numbers.
This is a broad category that encompasses all possible numbers without imaginary components.

• Rational numbers, such as fractions and decimals, can be expressed as a fraction \( \frac{a}{b} \) where \(a\) and \(b\) are integers, and \(b eq 0\).
• Irrational numbers are numbers that cannot be expressed as a simple fraction, like \( \pi \) or \( \sqrt{2} \).

Linear functions like \( h(x) = -8x - 2 \) accept any real number input for \(x\), since there are no operations that limit \(x\) to a specific set of numbers.
Hence, the domain of a linear function generally includes all real numbers because any real number can be plugged in for \(x\) without causing any mathematical issues.

Interval Notation

Interval notation is a shorthand used to express the domain or range of a function in mathematics.
It uses brackets to define the set of numbers that are part of the domain.

• Round brackets \(( )\) indicate that a number is not included in the interval (open interval).
• Square brackets \([ ]\) indicate that a number is included in the interval (closed interval).
• Infinity symbols \( \infty \) or \( -\infty \) are always used with round brackets because infinity is not a number that can be reached or counted.

For the function \( h(x) = -8x - 2 \), the domain expressed using interval notation is \((-\infty, \infty)\).
This tells us that every real number is a valid input for \(x\). Interval notation is an efficient way to convey this continuous range.