91Ó°ÊÓ

The square root function is a fundamental concept in mathematics, often written as \(f(x) = \sqrt{x}\). It asks, "What number, when multiplied by itself, gives \(x\)?" In other words, if \(y = \sqrt{x}\), then \(y^2 = x\). For example, some common square roots are:

• \(\sqrt{9} = 3\) because \(3 \times 3 = 9\)
• \(\sqrt{16} = 4\) because \(4 \times 4 = 16\)

The square root function is only defined for non-negative numbers, as you can’t take the square root of a negative number (without venturing into complex numbers). This function has a unique characteristic: it grows slower as \(x\) becomes larger. This means the increase in \(f(x)\) values becomes less significant as \(x\) increases. Knowing how to manipulate this function will help you calculate other important mathematical concepts, such as average rate of change.

The substitution method is a common technique used to simplify mathematical problems, often involving functions. It's about replacing a variable with a specific value. This method comes in handy when dealing with expressions or equations you need to evaluate. In our exercise, we used the substitution method to calculate \(f(x_{1})\) and \(f(x_{2})\). Here’s how:

• Replace \(x\) with \(x_{1} = 9\) in the function \(f(x) = \sqrt{x}\). This gives us \(f(x_{1}) = \sqrt{9} = 3\).
• Similarly, replace \(x\) with \(x_{2} = 16\) to get \(f(x_{2}) = \sqrt{16} = 4\).

Substitution is a powerful tool because it allows you to directly apply known values to functions, simplifying the process of finding solutions to complex problems.

The difference quotient is a fundamental concept in calculus that helps measure how a function changes. It is used to calculate the average rate of change of a function over a specified interval. The formula is:\[\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\]This represents the change in the function's output over the change in input between two points, \(x_{1}\) and \(x_{2}\).

• In the context of our problem, we have \(f(x_{2}) = 4\) and \(f(x_{1}) = 3\), with \(x_{1} = 9\) and \(x_{2} = 16\).
• Plug these values into the formula to get the difference quotient: \(\frac{4 - 3}{16 - 9} = \frac{1}{7}\).

The difference quotient gives us an important average rate of change for the function \(\sqrt{x}\) over the interval from 9 to 16. Understanding this concept is crucial for exploring derivatives and how functions behave over different inputs.

Interval notation is a mathematical way to describe a range of values. It efficiently communicates where a function is being evaluated. Understanding interval notation is essential as it simplifies specifying intervals.In interval notation, parentheses \(( )\) or brackets \([ ]\) are used:

• Parentheses \(( )\) indicate that an endpoint is not included.
• Brackets \([ ]\) indicate that an endpoint is included.

For example, the notation \((9, 16)\) specifies an interval from 9 to 16, excluding the endpoints. Whereas, \([9, 16]\) means the interval includes both endpoints.In our exercise, the interval \([9, 16]\) was important because it defined over which our rate of change was calculated. This notation helps in conveying mathematical ideas clearly and is a foundational skill often used in analysis and calculus. By mastering interval notation, you can better understand where functions apply and how they behave over defined ranges.