Problem 50 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then sketch its graph and find the range. $$y=2 \sqrt{x}+3$$

Step-by-Step Solution

Answer

The domain of the function \( y=2 \sqrt{x}+3 \) is \( x \geq 0 \), the graph is a shift of the graph of the square root function \( y = \sqrt{x} \) three units up and stretching vertically by a factor of 2 and the range of the function is \( y \geq 3 \).

Step 1: Find the Domain

The domain of the function consists of all possible x-values. Since this function contains a square root, and you can't take the square root of a negative number, the function is only defined for x-values that are greater than or equal to zero. So, the domain of the function is \( x \geq 0 \).

Step 2: Sketch the Graph

To sketch the graph, plot a few points on the graph and connect them. For instance, if \( x = 0 \), \( y = 2 \sqrt{0} + 3 = 3 \), which provides the point (0,3). If \( x = 1 \), \( y = 2 \sqrt{1} + 3 = 5 \), which provides the point (1,5). We can observe that this function is a shift of the graph of the square root function \( y = \sqrt{x} \) three units up and stretching vertically by a factor of 2

Step 3: Find the Range

Looking at the graph, we can notice that the lowest point on the graph is at y = 3. There is no upper boundary for y. The y-values on the graph are thus greater than or equal to 3. That means the range of this function is \( y \geq 3 \).

Key Concepts

Functions

Functions play a crucial role in mathematics, serving as a bridge between sets of numbers or objects. In the context of this exercise, we focus on a specific function: \[ y = 2 \sqrt{x} + 3 \] This function maps each input \( x \) to an output \( y \). The defining feature of a function is that for each value of \( x \), there is exactly one corresponding value of \( y \). In this function, the operations applied to \( x \) include taking the square root of \( x \), multiplying by 2, and then adding 3. It is important to understand both the domain and range of a function:- **Domain:** The set of all possible inputs \( x \) for which the function is defined.- **Range:** The set of all possible outputs \( y \) that the function can produce.For our function, since it involves a square root, the domain is limited to non-negative values of \( x \), specifically \( x \geq 0 \). The range, as determined after sketching the graph and examining the values \( y \) can take, is \( y \geq 3 \). Understanding these sets helps in visualizing and interpreting the behavior of functions.

Square Root

The square root symbol, \( \sqrt{} \), indicates the operation of finding a number that, when multiplied by itself, results in the original number under the radical. A key aspect of square roots in functions like \( f(x) = 2 \sqrt{x} + 3 \) is that the domain must be restricted to non-negative values. This is because the square root of a negative number is not defined within the real number system. Therefore, for this function:- Every \( x \geq 0 \) can be used.The effect of the square root operation is a gradual increase rather than a linear increase in the values of \( y \) as \( x \) increases. For instance:- When \( x = 0 \), \( y = 3 \)- When \( x = 1 \), \( y = 5 \)- When \( x = 4 \), \( y = 7 \)This demonstrates how the function values start at a minimum \( y = 3 \) and extend upwards as \( x \) increases. Understanding the square root component is essential for grasping how the function's graph behaves.

Graph Sketching

When sketching the graph of a function, visual observation becomes a powerful tool for interpreting mathematical relationships. For the function \( y = 2 \sqrt{x} + 3 \), graph sketching starts by identifying key points calculated from the function, such as:- \((0,3)\)- \((1,5)\)- \((4,7)\)These points are plotted on a coordinate plane, and a smooth curve is drawn through them. Since the function is a transformation of the basic square root function \( \sqrt{x} \), there's a vertical stretch by a factor of 2 and a translation up by 3 units.Visualizing these transformations helps in:- Recognizing shifts and stretches from the original \( \sqrt{x} \) graph.- Understanding the boundaries and behavior of the graph, particularly the domain \( x \geq 0 \) and range \( y \geq 3 \).This approach not only aids in completing exercises about specific functions but also builds a deeper understanding of how to interpret changes in the graph with respect to operations applied within the function itself.