Problem 51 - Algebra 1

Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=\sqrt{x-4}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\sqrt{x-4}\) is [4, \(+\infty\)) and the range is [0, \(+\infty\)). The graph starts from the point (4,0) and increases as \(x\) increases.

Step 1 - Identify the Domain

Set the radicand \(x-4\) greater than or equal to zero. To solve this inequality, add 4 to both sides to find \(x \geq 4\). Hence, the domain is all real numbers \(x\) such that \(x \geq 4\). It can also be denoted as [4, \(+\infty\)).

Step 2 - Identify the Range

Since the square root function outputs non-negative values and there's no operation altering the output, the function \(y\) will be always greater than or equal to zero. Thus, the range of the function is [0, \(+\infty\)).

Step 3 - Sketching the Graph

Start by choosing a number in the domain of \(x\), substitute it in the function to find the corresponding \(y\) value. Repeat the process to find other points. Connect these points to form the graph. The graph begins at point (4,0) and increases as the value of \(x\) increases.

Key Concepts

Understanding the Square Root Function

The square root function is a mathematical operation that returns the value which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. In the function provided in the exercise, \( y = \sqrt{x - 4} \), the square root is taken of the expression \( x - 4 \).

This function is defined only for \( x \) values where \( x - 4 \) is non-negative because square roots of negative numbers are not real. Therefore, the domain is the set of real numbers \( x \) such that \( x \geq 4 \), often written as \( [4, +\infty) \).

Since the output of a square root function is always non-negative, the range is from 0 to infinity, represented as \( [0, +\infty) \). It’s important to note that the graph of a square root function features a characteristic curve that starts at the point \((4, 0)\) for our specific function and continues to rise, approaching but never touching the vertical line \( x = 4 \) known as the vertical asymptote.

Inequalities in Algebra

Inequalities are a central part of algebra that describe the relative size or order of two values. They use symbols like \( > \), \( < \), \( \geq \), and \( \leq \) to show if one number is greater than, less than, greater than or equal to, or less than or equal to another number, respectively.

When you encounter a function like \( y = \sqrt{x - 4} \), setting the radicand (the expression inside the square root, in this case, \( x-4 \)) greater than or equal to zero creates an inequality, which is solved to find the domain of the function. This means that the number under the square root, \( x - 4 \), must be greater than or equal to 0; thus, \( x \) must be at least 4.

Understanding inequalities helps in sketching graphs and determining the domain and range of functions, as it clearly establishes the possible values that variables can take, ensuring the function provides real number outputs.

Graphing Functions

Graphing is a powerful tool in algebra that allows you to visually interpret the behavior of functions. For the given function \( y = \sqrt{x - 4} \), we first determine its domain and range. The graph of this function starts at the point \((4, 0)\) and, as \( x \) increases, the value of \( y \) also increases, creating a curve that moves upwards to the right.

To sketch the graph, choose several values of \( x \) within the domain and compute the corresponding \( y \) values. These points can then be plotted on a Cartesian plane. Connecting these points in a smooth curve illustrates the function's behavior graphically. The curve of the square root function is gradual; it grows slower as \( x \) increases due to the nature of square root growth, which is slower compared to linear or polynomial functions. Remember, the graph will not extend to the left of the \( y \)-axis since the function is not defined for \( x < 4 \).