91Ó°ÊÓ

When we talk about the domain and range of a function, we're essentially discussing where the function "lives" on a graph. For the quadratic function \(f(x) = x^2\), with the restriction \(-2 \leq x \leq 5\), the **domain** refers to all possible x-values that we can plug into the function. You can picture this as the function "living" on the x-axis from \(-2\) to \(5\). So, in this scenario, we describe the domain using interval notation as \([-2, 5]\).

On the other hand, the **range** is concerned with the y-values that result from evaluating the function at every x-value in its domain. For this function, the smallest y-value is when \(x=0\) (which is \(f(0)=0\)), and the largest is \(f(5)=25\). Hence, the range is noted as \([0, 25]\). These include all numbers from \(0\) to \(25\) that result from squaring numbers between \(-2\) and \(5\). Utilize interval notation to clearly communicate your domain and range.

A graphing calculator is an incredibly useful tool for visualizing mathematical functions. In this exercise, we use it to graph the function \(f(x) = x^2\). To successfully plot this function, input the equation into the calculator and ensure the x-values are limited to the specified restriction \(-2 \leq x \leq 5\).

Here's why a graphing calculator is great:

• It helps you visualize functions, making it easier to understand their shape and behavior.
• You can quickly modify the interval variables, which aids in seeing how a function behaves over different ranges.
• It provides immediate feedback, which is invaluable for learning and exploration.

For this exercise, confirm that the graph represents a section of a parabola that opens upwards, displaying a clear vertex at the origin.

A **parabola** is a U-shaped curve that represents the graph of a quadratic function like \(f(x) = x^2\). In our function, since the x-values range from \(-2\) to \(5\), we get a partial parabola rather than the full infinite curve you might usually see.

Here are a few essential characteristics of parabolas:

• A parabola has a **vertex**, which is its highest or lowest point depending on the direction it opens. For our function, the vertex is at \((0,0)\).
• The direction of opening (upward or downward) is determined by the sign of the coefficient in front of \(x^2\). Here it's positive, so our parabola opens upward.
• In our exercise, the parabola starts rising at \(x = -2\), hits its lowest point at \(x = 0\), and continues rising until \(x = 5\).

By understanding the properties of parabolas, you gain better insights into how quadratic functions behave over their domain.

**Interval notation** is a convenient and concise way to describe a set of numbers. It helps in clearly articulating the domain and range of functions in a mathematical context.

Here's how interval notation works:

• Square brackets \([ ]\) indicate that an endpoint is included, known as *inclusive*.
• Parentheses \(( )\) would indicate an endpoint is excluded, called *exclusive*. However, for our problem, the endpoints are inclusive.

So when we say the domain of \(f(x) = x^2\) is \([-2, 5]\), it means x-values start inclusive of \(-2\) and end inclusive of \(5\). Similarly, describing the range as \([0, 25]\) conveys that the y-values start at \(0\) and end at \(25\), both included. Interval notation is an elegant way of writing continuous values in mathematics.