Problem 22 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$g(x)=\sqrt{x}+5$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \sqrt{x} + 5\) is [0, +∞), and the range is [5, +∞).

Step 1: Identify The Function Type

In this case, the function \(g(x) = \sqrt{x} + 5\) is a transformation of the parent function \(\sqrt{x}\). It is shifted upwards by 5 units.

Step 2: Determine the Domain of the Function

The square root function \(\sqrt{x}\) is defined only for x greater than or equal to zero. Thus the domain of \(\sqrt{x} + 5\) is also all x greater than or equal to zero, conveying that we can only substitute real numbers that are greater than or equal to zero into the function. So the domain is [0, +∞).

Step 3: Determine the Range of the Function

Let's examine the range of \(g(x) = \sqrt{x} + 5\). The smallest value for \(\sqrt{x}\) is 0 , when x is 0. The 5 added to the square root then shifts all the y-values 5 units up. Hence, the minimum value of \(g(x)\) is 5. Therefore, the range of the function is [5, +∞).

Key Concepts

Function Transformation

When we talk about function transformation, we refer to the ways in which a basic function, also known as the 'parent function', can be altered. These changes can include shifting, stretching or compressing, and reflecting the graph of a function.
One of the most common transformations is vertical shifts, which involve moving the entire graph of the function up or down. In the function \(g(x) = \sqrt{x} + 5\), the parent function \(\sqrt{x}\) is vertically shifted up by 5 units.
This means that every point on the parent function, represented by \(x\), is moved 5 units higher in the y-direction.
This is a simple way to modify a function without altering its general shape. It's important to grasp these transformations to predict how a function's graph will change and how its domain and range are affected.

Square Root Function

The square root function, \(f(x) = \sqrt{x}\), is one of the elementary functions in mathematics and is widely understood for its unique properties. One key characteristic of \(f(x) = \sqrt{x}\) is that it's only defined for non-negative numbers, meaning the domain is \([0, +\infty)\). This is because there is no real square root for negative numbers.
Additionally, since the smallest value \(\sqrt{x}\) can take is 0, the range of the parent square root function is also \([0, +\infty)\).
The function begins at the origin point \((0,0)\) and increases slowly. This gradual increase is displayed on a graph as a curve that flattens out as \(x\) gets larger. Understanding these aspects of the square root function helps with recognizing and solving transformations, like adding a constant value to shift the graph.

Parent Function

A parent function is the simplest form of a family of functions that maintains the general shape of that family. Here, \(\sqrt{x}\) is the parent function for radical functions, and it represents the most basic form of a square root graph. The parent function is a starting point for understanding transformations that yield new functions.
This foundational graph for the function \(\sqrt{x}\) serves as a reference when transformations are applied, such as shifts, reflections, or stretches.
The primary characteristics, such as domain, range, and the curve's general shape, are pivotal when we look at how more complex functions can be derived. By identifying and understanding parent functions, solving transformations and interpreting changes in the domain and range become intuitive and straightforward, giving students a solid basis for tackling more complicated problems.