Problem 54 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then sketch its graph and find the range. $$y=\sqrt{3 x+1}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\sqrt{3 x+1}\) is \([-1/3, \infty)\), the range is \([0, \infty)\), and the graph is a sideways parabola starting at point (-1/3, 0) and increasing as x increases.

Step 1: Find the Domain

In order to find the domain of the function, which is the set of all possible x-values, set the expression inside the square root \(3 x+1\) to be greater than or equal to 0: \[3 x+1 \geq 0 \]Solving this, one gets \(x \geq -1/3\). So the domain of the function \(y=\sqrt{3 x+1}\) is \([-1/3, \infty)\)

Step 2: Sketch the Graph

The square root function starts from -1/3 (due to the domain derived from step 1) and continue upwards, as the square root of a number is positive. This graph represents half of a parabola turned on its side. At x = -1/3, y = 0; as values for x increase, y also increases. You can plot the points (-1/3, 0), (0, 1) and (1, 2) to get the general shape of the graph.

Step 3: Find the Range

The range of a function is the set of all possible values of the output (y). This function will output all y-values greater than or equal to zero, since x is greater than or equal to -1/3. Hence, the range of the function is \([0, \infty)\)

Key Concepts

range of a function

In mathematics, the range of a function is a critical concept that refers to all possible output values (y-values) a function can produce. To understand the range of the function \(y=\sqrt{3x+1}\), we need to examine how the values of \(x\) affect the values of \(y\).

Here's how you can determine the range:

• Firstly, consider the operation involved: a square root. By definition, the square root function \(\sqrt{a}\) is only defined for \(a \geq 0\), and its outputs are non-negative.
• For \(y=\sqrt{3x+1}\), when \(x\) is at its minimum value of \(-1/3\), we substitute into the equation to find \(y=0\).
• As \(x\) increases beyond \(-1/3\), \(3x+1\) becomes positive and larger, making the square root increase further.

Thus, the smallest value for \(y\) is 0, and it increases without bound as \(x\) increases. Therefore, the range of this function is \([0, \infty)\). This means \(y\) will be zero or any positive number as \(x\) varies within its domain.

graphing functions

Graphing functions is a visual representation of how a function behaves. For the function \(y=\sqrt{3x+1}\), graphing it helps understand its domain and range better.

Here are some steps to make the graph:

• Identify the domain: We already found that \(x\) must be greater than or equal to \(-1/3\). Hence, our graph only exists for these x-values.
• Find key points: Calculate a few values to plot on the graph for clarity. When \(x = -1/3\), \(y = 0\). When \(x = 0\), \(y = 1\), and for \(x = 1\), \(y = \sqrt{3}\approx 1.73\).
• Plot these points on a coordinate plane.

Now, join the points smoothly, ensuring the curve reflects the square root form; it should start at \(x = -1/3\) and move up to the right. This function's graph will look like half of a parabola, opening to the right, due to the square root shape. Remember, the graph never falls below the x-axis, reflecting the range \([0, \infty)\).

square root functions

Square root functions are a type of radical function, with the square root symbol \(\sqrt{ }\), and they often present unique graph shapes and characteristics.

Understanding their behavior involves:

• This function, \(y=\sqrt{3x+1}\), has the expression inside the square root \(3x+1\). The square root requires this to be non-negative, which defines the domain \(x\geq -1/3\).
• The output is always non-negative because the square root of any real number is positive, covering the range \([0, \infty)\).
• Graphically, these functions start from a point depending on the domain and rise steadily, never decreasing, as the value under the root increases.

Square root functions are continuous where defined and represent specific, predictable growth patterns. These functions can be transformed further to capture different slopes or vertical shifts, but follow the same basic shape. Understanding them is essential across various scientific applications, as they frequently model real-world phenomena like idealized growth processes.