Problem 18 - Precalculus Essentials

Question

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$f(x)=\sqrt{x} \text { from } x_{1}=9 \text { to } x_{2}=16$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x)=\sqrt{x}\) from \(x_1 = 9\) to \(x_2 = 16\) is \(\frac{1}{7}\).

Step 1: Identify Function

Identify the function given which is \(f(x)=\sqrt{x}\) and the points \(x_1 = 9\) and \(x_2 = 16\).

Step 2: Substitute into the Function

Now we substitute those values into our function. So, \(f(x_1) = \sqrt{9} = 3\) and \(f(x_2) = \sqrt{16} = 4\).

Step 3: Apply the Rate of Change Formula

Calculate the average rate of change of the function. Substitute all the values in the formula \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\). \(\frac{4-3}{16-9}= \frac{1}{7}\)

Key Concepts

Functions in Precalculus

When studying functions in precalculus, it's crucial to understand that a function denotes a special relationship between two sets: the domain and the range. In simple terms, it pairs each input from the domain with exactly one output in the range.

For example, the square root function, which we'll explore further, maps non-negative numbers (its domain) to their square root values (its range). This one-to-one correspondence between ordered pairs can be represented visually on a graph, with the input values along the x-axis and the output values along the y-axis.

A function like the square root function, which involves taking the square root of the input value, is a fundamental type of function known for its curved graph that slowly levels off as the input values increase. Students should be comfortable with evaluating such functions at specific points and visualizing their graphs to understand their behavior.

Rate of Change Formula

The rate of change formula is essential in understanding how a function behaves between two points. Specifically, the average rate of change gives us the slope of the secant line that goes through the points \( \(x_1, f(x_1)\) \) and \( \(x_2, f(x_2)\) \). It's calculated as \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).

If the function represents something like speed over time, the average rate of change is the average speed within that interval. In other contexts, this concept could represent the average rate at which costs increase or decrease, the growth rate of plants, and many other real-world situations.

To make this concept more tangible for students, consider two points on a mountain. The average rate of change in this scenario could represent the overall steepness of the incline or decline between those two points. This same principle applies in mathematics to represent the steepness or the average rate at which the function values (or 'heights') are changing between two input values.

Square Root Functions

Square root functions are a special subset of functions characterized by the presence of a square root in their expression, often written in the form \( f(x) = \sqrt{x} \). They're vital for solving equations where the variable is under a square root and have distinctive graphs that take the shape of a half-parabola lying on its side.

For the square root function, the domain is all non-negative numbers, since square roots of negative numbers aren't defined in the set of real numbers. The range is also non-negative because the square root of any non-negative number is always non-negative.

When analyzing the average rate of change for a square root function over an interval, you can think of it as how quickly the 'height' of the function is changing as we move along the 'ground' from one point to another. The average rate of change for these functions will decrease as the input value gets larger, which can be observed from the graph flattening out as \(x\) increases.