Problem 50 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The domain of the function is \([0, +\infty)\) and the range is \([-3, +\infty)\). The graph starts at the point (0,-3) and increases gradually as x increases but never drops below -3.

Step 1: Identify the Domain

Since we are dealing with a square root function, and the value under the square root, (x), must be greater than or equal to 0, the domain can be written as \(x \geq 0\). In interval notation, the domain is \([0, +\infty)\)

Step 2: Determine the Range

The range of the function is determined by the possible output values. In this function, every output value will be the square root of x, subtracted by 3. As x>=0 and the smallest possible value for \( \sqrt{x} \) will be 0 (when x=0), the smallest possible value for y will be \( \sqrt{0} - 3 = -3 \). Therefore, the range of the function is Y: \([-3, +\infty)\)

Step 3: Sketch the Graph

The graph of the function \(y = \sqrt{x} - 3\) is a horizontal shift of the graph of \(y = \sqrt{x}\) 3 units down. This means the graph starts at the point (0,-3), and as x moves away from 0, y will increase. This results in a graph that gradually increases but never drops below -3 as x increases.

Key Concepts

Understanding the Domain of a Function

The domain of a function describes all possible input values (typically represented by the variable x) for which the function is defined. In essence, it tells us what x-values we can put into our function without causing any problems, like division by zero or taking the square root of a negative number.

For square root functions, such as the one given by \( y = \sqrt{x} - 3 \), the expression inside the square root, which is x in this case, must be non-negative (greater than or equal to zero).

• This is because the square root of a negative number is not considered a real number in basic algebra.
• Therefore, the domain of this specific function is all x-values starting from 0 and extending to infinity.
• We express this in interval notation as \([0, +\infty)\).

By understanding the domain, we know which values are acceptable for x, ensuring we avoid undefined mathematical operations.

Exploring the Range of a Function

The range of a function refers to all possible output values, or y-values, that result from using the domain. In simple terms, it defines what values the function can actually take.

For the function \( y = \sqrt{x} - 3 \), the range depends on the outputs of the square root term before subtraction.

• Initially, the smallest value \( \sqrt{x} \) can be is 0, which occurs when \( x = 0 \).
• Thus, when \( \sqrt{x} = 0 \), the result of the function \( y = 0 - 3 \), giving a minimum y-value of -3.
• Since \( \sqrt{x} \) can take any non-negative value, subtracting 3 just shifts all values down.
• Hence, as \( x \) increases, y will continue to grow without bound above -3.

The range is therefore represented in interval notation as \([-3, +\infty)\), starting from the lowest output value and moving upwards indefinitely.

Insights into Square Root Functions

Square root functions are a specific type of function where the variable x is found inside a square root.
These functions have unique characteristics which are important to understand.

• The basic structure of a square root function is \( y = \sqrt{x} \).
• This basic version always starts at the origin (0,0) and curves upwards to the right.
• Additions or subtractions outside the square root, like \(-3\) in \( y = \sqrt{x} - 3 \), will shift the graph up or down.

In the case of \( y = \sqrt{x} - 3 \), every point on the basic square root curve is adjusted downward by 3 units. This means the function begins at (0, -3) on a graph.

Square root functions are common in algebra because they inherently model scenarios where output values increase gradually. They generate a curve that increases rapidly at first and then grows more slowly as x becomes larger. This makes understanding both the domain and range key to accurately interpreting their behavior on a graph.