Problem 56 - Algebra 1: Concepts and Skills

Question

In Exercises 54–56, use the function y x 9, where 2 ? x ? 6. (Lesson 1.8) State the domain and range of the function.

Step-by-Step Solution

Answer

The domain of the function is \([2, 6]\) and the range is \([512, 10077696]\).

Step 1: Review the function

The function is \(y = x^9\), where \(2 \leq x \leq 6\). It is a polynomial function with only positive powers of x, as a result it is defined for all real numbers.

Step 2: Check the domain

The domain of the function \(y = x^9\) has been specified as \(2 \leq x \leq 6\). So, the domain is the interval \([2, 6]\) in the real number line.

Step 3: Calculate the range

To calculate the range, substitute the boundary values of the domain into the function. For \(x=2\), \(y = (2)^9 = 512\). For \(x=6\), \(y = (6)^9 = 10077696\). So, the range of the function is \([512, 10077696]\).

Key Concepts

Polynomial Functions

Polynomial functions form a foundational concept in algebra and calculus. They are expressed as sums of terms consisting of a variable raised to a non-negative integer power, like the function presented in the given exercise, which is defined as \(y = x^9\). This specific function is notable because it involves a single term where the variable 'x' is raised to the ninth power, reflecting a basic form of a polynomial called a monomial.

One of the remarkable properties of polynomial functions like this is their continuity and smoothness over the entire real number line. This means that for any input within the domain of the function, there is a specific output, which contributes to the ease of determining values for both domain and range. For students, understanding polynomial functions is crucial because they model a wide range of real-world phenomena and are the stepping stones to more complex functions in advanced mathematics.

Interval Notation

Interval notation is a system of representing subsets of the real number line using brackets and parentheses to communicate the starting and ending points of these intervals. In the context of functions, it is often used to express the domain and range. For example, the exercise demonstrates the domain of the polynomial function \(y = x^9\) with the notation \([2, 6]\).

This tells us that 'x' starts at 2 and ends at 6, and both endpoints are included in the domain, indicated by the use of square brackets. If an endpoint were not included, a parenthesis would replace the bracket. This notation is not only concise but also universally understood in mathematical discourse, and it's essential for students to be fluent in interval notation to effectively communicate the scope of mathematical concepts.

Real Number Line

The real number line is the visualization of the set of all real numbers. Any point on this line represents a real number, making it an invaluable tool in understanding concepts such as the domain and range of functions. When we speak of the domain of the polynomial function \(y = x^9\), determining that \(2 \leq x \leq 6\) places a segment on the real number line between the numbers 2 and 6.

Similarly, when the exercise asks us to find the range, we use the real number line to observe outputs of the function for inputs within the specified domain. This helps in visualizing the spread of the function's values and is especially useful when dealing with complex functions where manual calculation of the range would be less practical. Students should grasp the concept of the real number line as it is central not only to plotting functions but also to understanding number sets, and operations for calculus.