Problem 61 - Algebra 1

Question

Use the equations \(y=\frac{1}{x}\) and \(y=\frac{1}{x+3}\). How can you restrict the domain of each equation so that it is a function?

Step-by-Step Solution

Answer

For the function \(y=\frac{1}{x}\), it is defined for all real numbers except 0. And for the function \(y=\frac{1}{x+3}\), it is defined for all real numbers except -3. Hence, the domain of the first function is all real numbers except 0 i.e. (-∞, 0) union (0, ∞) and the domain of the second function is all real numbers except -3 i.e. (-∞, -3) union (-3, ∞).

Step 1: Identify the restriction for \(y = \frac{1}{x}\)

If we set the denominator equal to zero, we get it for x=0. So, for \(y = \frac{1}{x}\), the domain is all real numbers except 0, which can be written as (-∞, 0) union (0, ∞). That is the set of all real x such that x ≠ 0.

Step 2: Identify the restriction for \( y = \frac{1}{x+3}\)

Similarly, for defining the domain of \(y = \frac{1}{x+3}\), we set the denominator equal to zero and solve for x, to get x=-3. Hence, for \(y = \frac{1}{x+3}\), the domain is all real numbers except -3, which can be written as (-∞, -3) union (-3, ∞). That is the set of all real x such that x ≠ -3.

Key Concepts

Rational Functions

Rational functions are mathematical expressions defined by a ratio of two polynomials. In simpler terms, they are fractions where both the numerator and the denominator are polynomials. An example is the function \( y = \frac{1}{x} \) where the numerator is 1 (which is a polynomial, despite being just a constant), and the denominator is the polynomial \( x \).

One of the critical properties of rational functions is that they are undefined when their denominator is zero since division by zero is not allowed in mathematics. For the function \( y = \frac{1}{x} \) the value of \( x \) cannot be 0, as it would make the denominator zero. Hence, when identifying the domain - which is the set of all possible input values (often \( x \) values) - we must exclude the value which causes the denominator to be zero.

Function Restrictions

The domain of a function is sometimes constrained or 'restricted' due to its mathematical properties. These restrictions prevent the function from being undefined or unmanageable within certain contexts. For example, the function \( y = \frac{1}{x+3} \) cannot take the value \( x = -3 \) because it would result in division by zero.

To articulate these restrictions, we often use interval notation or set-builder notation. For instance, we express the domain of the function \( y = \frac{1}{x} \) as \( (-\infty, 0) \cup (0, +\infty) \), signifying that \( x \) can be any real number except 0. This restriction ensures that the output of the function remains defined and the relationship between the input and output remains consistent as per the function's definition.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes all the rational and irrational numbers. Rational numbers are the ones that can be expressed as a fraction of two integers, while irrational numbers are non-repeating, non-terminating decimals.

When discussing domains of functions, especially rational functions, it is vital to consider the set of real numbers because the domain is fundamentally the set of all possible input values that are real numbers. Any constraints or exclusions to the domain, such as those due to division by zero, must be expressed in terms of the real numbers that are not included in the function.