Problem 16 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then sketch its graph. $$y=\sqrt{x}+5$$

Step-by-Step Solution

Answer

The domain of the function \(y = \sqrt{x} + 5\) is \(x \geq 0\). The graph starts from the point (0,5) and moves upward.

Step 1: Define the function

The first step is to define the function, which is \(y = \sqrt{x} + 5\).

Step 2: Determine the domain of the function

Let's inspect the square root function to determine the domain. Remember, x must be nonnegative for the square root to be a real number. Thus, the domain will be \(x \geq 0\).

Step 3: Sketch the graph of the function

Now we can sketch the graph starting from the point (0,5). The graph will move upwards starting from this point because for every x, the y value increases.

Key Concepts

Square Root Function

Understanding the square root function is essential for grasping various mathematical concepts. The function is denoted as \( y = \sqrt{x} \) and involves taking the square root of the input variable \( x \). A key characteristic of square root functions is that they are defined only for non-negative values of \( x \), since the square root of a negative number is not a real number, but an imaginary one. This restriction is what dictates the domain of the function.

In the context of our exercise, the function \( y = \sqrt{x} + 5 \) is a square root function that has been vertically shifted upwards by 5 units. This transformation affects the graph but not the domain. The domain remains all real numbers \( x \geq 0 \) because the square root of a negative number is undefined in the set of real numbers. Thus, when trying to find the domain of any square root function, you should look for all the values of \( x \) where the expression under the square root is non-negative.

Graphing Functions

Graphing functions is a pivotal skill for visualizing mathematical relationships. The graph of a function represents all the possible pairs of input and output values, plotted as points in the Cartesian plane. To graph a function like \( y = \sqrt{x} + 5 \) effectively, we must understand the shape of its parent function \( y = \sqrt{x} \) and how it changes with any additional transformations, such as vertical shifts, reflections, or stretches.

Starting with the parent function \( y = \sqrt{x} \)—it has a characteristic 'half-parabola' shape, opening to the right, and starting from the origin (0,0). When graphing \( y = \sqrt{x} + 5 \) we begin by shifting this curve up by 5 units. The resulting graph still increases as \( x \) increases, but now it starts at the point (0,5). It’s important to note that even though we shift the graph upwards, it does not alter the domain; it only affects the vertical placement of the graph, which in turn changes the range of the function. Remember to draw the graph smoothly and progressively as the values of \( y \) increase with increasing \( x \) while respecting the predefined domain.

Real Numbers

The concept of real numbers is fundamental in understanding the domain of a function. Real numbers include all the numbers on the number line, comprising rational numbers (like 3, -1, 0.5) and irrational numbers (like \( \sqrt{2} \) or \( \pi \) ). They can be positive, negative, or zero. When discussing the domain of a function, we're essentially asking which real numbers can be plugged in for \( x \) to give a real-number output.

In terms of the square root function, only non-negative real numbers are allowed because the square root is not defined for negative numbers within the real number system. This restriction creates a domain of \( [0, \infty) \) for our function \( y = \sqrt{x} + 5 \) which means any real number \( x \geq 0 \) is an acceptable input. When solving for domains, always consider the type of numbers needed to make the function work—whether they must be non-negative, as with square roots, or perhaps within a certain interval or set of values to avoid undefined expressions.