Problem 47 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The domain of \(y=\sqrt{x}-3\) is \(x \geq 0\) and the range is \(y \geq -3\). The graph starts at the point (0,-3) and then increases gradually.

Step 1: Find the Domain

The domain of the given function is all possible inputs values which, in earth terms, is the set of all x-values that you are allowed to use. The square root function only takes non-negative values. Hence the domain of \(y=\sqrt{x}-3\) will be \(x \geq 0\).

Step 2: Find the Range

The range of a function is the set of all possible output values (y-values). Looking at the function \(y=\sqrt{x}-3\), the smallest value of \(\sqrt{x}\) would be zero (when \(x=0\)), which would make \(y=-3\). As \(x\) increases, so does \(\sqrt{x}\) and this in turn, will satisfy all \(y\) values which are greater than -3. Hence, the range of \(y=\sqrt{x}-3\) is \(y \geq -3\).

Step 3: Sketch the Graph

The graph of the function \(y=\sqrt{x}-3\) will be the graph of \(y=\sqrt{x}\) shifted 3 units down along the y-axis. The graph starts at the point (0,-3) and then increases gradually.

Key Concepts

Understanding the Function Graph

A function graph visually represents the relationship between inputs and outputs of a function. Each point on the graph corresponds to a pair \((x, y)\), where \(y\) is calculated from \(x\). This visual tool helps to grasp how a function behaves across its domain.

For the function \(y = \sqrt{x} - 3\), the graph will reveal how the output \(y\) changes as \(x\) increases. The domain \(x \geq 0\) means that the graph will only exist for non-negative \(x\) values, originating from the starting point \((0, -3)\), as calculated earlier. This depicts the smallest \(y\) in the range, due to the smallest possible value of \(\sqrt{x}\), which is zero. The slope of the graph is gentle and increases upward as \(x\) grows, since the square root function naturally increases slowly.

Thus, a function graph is not just about connecting dots; it displays patterns and potentially infinite sequences of coordinates that all align according to the rule defined by the function.

Characteristics of the Square Root Function

The square root function, \(y = \sqrt{x}\), is a fundamental non-linear function with its own set of behaviors and rules. Unlike linear functions which have a constant rate of change, the square root function grows at a decreasing rate, meaning it rises slower and slower.

• Non-negative Input: Only non-negative numbers have real square roots, hence, it restricts its domain.
• Always Increasing: As \(x\) increases, \(\sqrt{x}\) always increases, but at a decreasing rate.
• Starts at Origin: It starts at the point \((0,0)\) and continues on as \(x\) grows larger

In the modified function \(y = \sqrt{x} - 3\), this base function gets shifted downward, altering its range but maintaining its characteristic shape and rules. The domain remains \(x \geq 0\), but the range transforms to \(y \geq -3\), reflecting this simple yet profound mathematical behavior.

Exploring Function Transformations

Function transformations change a function's position, shape, or orientation. Transformations include translations, reflections, stretching, and compressing. They allow us to see how functions relate to each other by applying slight alterations.

In our function \(y = \sqrt{x} - 3\), a basic square root function \(y = \sqrt{x}\) is transformed by translating the whole graph 3 units downward. This operation is termed a vertical shift. The whole graph shifts lower along the y-axis without any change in its fundamental shape.

• Vertical Shift: Adjust the output by a constant. Here, it's shifted down by 3 units.
• Domain and Range Post-Transformation: These shifts directly impact the range, moving the smallest point from 0 to -3, as the whole graph adjusts the y-values accordingly.

Understanding how transformations work is essential to manipulating and predicting the behavior of various functions by adjusting parameters like this simple subtraction.