Problem 4 - Algebra 2

Question

Graph each function. Identify the domain and range. \(z(x)=-3\)

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \{-3\}; Graph is a horizontal line at \(y = -3\).

Step 1: Understand the Function

The given function is a constant function, specifically defined as \( z(x) = -3 \). This means that for any input \( x \), the output will always be \(-3\). There is no variable component to this function.

Step 2: Identify the Domain

The domain of a function is the set of all possible input values. For \( z(x) = -3 \), any real number can be an input because there are no restrictions on \( x \). Therefore, the domain is all real numbers, or \( (-\infty, \infty) \).

Step 3: Identify the Range

The range of a function is the set of all possible output values. Since \( z(x) = -3 \) is a constant function, the only output value is \(-3\). Thus, the range is simply \{-3\}.

Step 4: Graph the Function

To graph \( z(x) = -3 \), plot a horizontal line where the value of the function is \(-3\) across all values of \( x \). This means drawing a straight line parallel to the x-axis at \(-3\) on the y-axis.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually denoted as \(x\)) that can be fed into the function. In simpler terms, it's the complete set of x-values that will make the function work without any issues like dividing by zero or taking the square root of a negative number.
For the constant function \(z(x) = -3\), the situation is straightforward. There are no fractions to worry about (so no division by zero), and no square roots that could potentially lead to negative results. This means there's nothing stopping \(x\) from being any real number.
Thus, the domain of this function is all real numbers, which we write as \((-fty, fty)\). This wide domain makes constant functions very flexible and easy to work with in terms of inputs.

• The domain of \(z(x) = -3\) is all real numbers.
• Written in interval notation, the domain is \((-fty, fty)\).

Range of a Function

The range of a function is all the possible output values a function can produce. While the domain deals with inputs, the range deals with outputs. For constant functions like \(z(x) = -3\), determining the range is very simple.
No matter what value is chosen for \(x\), the output will always be \(-3\). This is because a constant function does not change; it remains the same for every \(x\). Thus, the range does not stretch over a series of values but is limited to a single value.
In the case of our function, the range is simply \{-3\}.

• The range indicates possible outputs of a function.
• For \(z(x) = -3\), the range is \{-3\}.

This makes constant functions very predictable and easy to analyze, as there is no variability in the outputs.

Graphing Functions

Graphing a function is all about visually representing the set of ordered pairs formed by inputs (x-values) and their corresponding outputs (y-values).
For a constant function like \(z(x) = -3\), this is both simple and convenient. You plot a horizontal line across the graph where the y-value is always \(-3\). This line is parallel to the x-axis, and each point on this line corresponds to an input value paired with the constant output of \(-3\).

• To graph \(z(x)=-3\), draw a straight line at \(y = -3\).
• This line runs horizontally and doesn’t change with different \(x\) values.

Using a graph provides a clear visual understanding of how a function behaves, especially useful for constant functions like this one, where the function's behavior is predictably unchanging.