Problem 52 - Intermediate Algebra

Question

Complete each table and then graph the function. Give the domain and range. See Examples 3 and 4. $$ f(x)=\sqrt{x}+2 $$ $$ \begin{array}{|r|r|} \hline x & y \\ \hline 0 & \\ 1 & \\ 4 & \\ 9 & \\ 16 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Domain: \( x \geq 0 \); Range: \( y \geq 2 \). Graph the curve using calculated points.

Step 1: Calculate Output Values

We need to find the value of the function \( f(x) = \sqrt{x} + 2 \) for each given \( x \). Calculate as follows:- For \( x = 0 \): \( f(0) = \sqrt{0} + 2 = 2 \).- For \( x = 1 \): \( f(1) = \sqrt{1} + 2 = 3 \).- For \( x = 4 \): \( f(4) = \sqrt{4} + 2 = 4 \).- For \( x = 9 \): \( f(9) = \sqrt{9} + 2 = 5 \).- For \( x = 16 \): \( f(16) = \sqrt{16} + 2 = 6 \).Fill in these values in the table.

Step 2: Complete the Table

After calculating the output values for \( f(x) = \sqrt{x} + 2 \), the table should look like:\[\begin{array}{|r|r|} \hline x & y \\hline 0 & 2 \1 & 3 \4 & 4 \9 & 5 \16 & 6 \\hline \end{array} \]

Step 3: Graph the Function

Plot the points from the completed table on a coordinate plane: \( (0, 2), (1, 3), (4, 4), (9, 5), (16, 6) \). These points should form a curve starting at \( (0, 2) \) and increasing, showing the general shape of the square root function shifted upwards by 2 units.

Step 4: Determine the Domain and Range

The domain of the function \( f(x) = \sqrt{x} + 2 \) is all non-negative real numbers, i.e., \( x \geq 0 \). The range is all real numbers \( y \geq 2 \), since the smallest value \( y \) achieves is \( 2 \) when \( x = 0 \).

Key Concepts

Understanding Domain and Range

The domain of a function is the set of all possible input values, typically represented by the variable \( x \). For the function \( f(x) = \sqrt{x} + 2 \), the domain consists of all non-negative real numbers. This is because the square root function, \( \sqrt{x} \), is only defined for \( x \geq 0 \). If \( x \) were negative, \( \sqrt{x} \) would be undefined within the realm of real numbers. Therefore, for our function, the domain is written as \( x \geq 0 \) or in interval notation as \([0, \infty)\).

Range, on the other hand, is the set of all possible output values. For \( f(x) = \sqrt{x} + 2 \), the smallest value \( f(x) \) can take is when \( x = 0 \), which results in a value of \( 2 \). As you increase \( x \), \( \sqrt{x} \) increases, thus \( \sqrt{x} + 2 \) also increases. Therefore, the range of this function is \( y \geq 2 \) or in interval notation, \([2, \infty)\).

Understanding domain and range is crucial as it defines the boundaries within which a function can be realistically applied and interpreted.

Graphing Functions

Graphing functions is like giving a visual representation to the relationship defined by the function's equation. To graph \( f(x) = \sqrt{x} + 2 \), you plot points that satisfy this equation on a coordinate plane. In this function, the table of values is a useful tool. By substituting specific \( x \)-values into the function, you calculate their corresponding \( y \)-values. These points are \((0, 2), (1, 3), (4, 4), (9, 5), \text{and} (16, 6)\).

Once you have these points, plot them on a coordinate grid. The graph should start at the point \( (0, 2) \) and form a smooth, increasing curve as \( x \) increases. This curve represents the square root function shifted upwards by 2, consistent with the transformation \( \sqrt{x} \to \sqrt{x} + 2 \).

Graphing does not only provide a clear picture of how \( y \) changes as \( x \) changes but also helps in verifying if the calculated points make sense in the context of the function. It's an effective way to validate your understanding of the function.

Square Root Functions

Square root functions have the general form \( f(x) = \sqrt{x} \). These functions are characterized by their unique curve that starts at a particular point and increases gradually. For \( f(x) = \sqrt{x} + 2 \), the function is derived from the basic square root function \( \sqrt{x} \). The expression \( + 2 \) indicates a vertical translation of the graph upwards by 2 units.

The behavior of square root functions is determined by the fact that they produce positive outputs for non-negative inputs. As you increase \( x \), the function value \( \sqrt{x} \) grows, but at a decreasing rate. This eventually gives the graph its characteristic curve.

Square root functions often arise in problems involving area calculations and physics, where square properties are essential. Understanding this function type is fundamental as it appears frequently in various mathematical and practical applications. Knowing how to graph, interpret, and manipulate these functions opens up a plethora of possibilities in problem-solving.