Question

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

Short answer

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

Step-by-step solution

- Find the domain

The function is \(y=\sqrt{x}+4\). Since a square root cannot have a negative number under the radical, x has to be greater than or equal to 0. Hence, the domain is \([0, +\infty)\). Any value we replace x with within the above interval will yield a real number when plugged into the function.

- Find the range

The smallest value \(\sqrt{x}\) can have is 0 (at x=0). Therefore, the smallest possible value for y is 4 (0+4). As we are adding 4 to \(\sqrt{x}\), the function becomes larger as x increases. Therefore, the largest possible value for y is infinity. Hence, the range is \([4, +\infty)\).

- Sketch the graph

Since the function is y= \(\sqrt{x}+4\), the graph will start at the point (0,4). As x increases, the function becomes larger. Therefore, the curve is upward sloping. The curve does not include any values below y=4, as according to the domain and range the curve lies in the first quadrant.

Key concepts

Square Root Function

Understanding the square root function is fundamental to grasp why certain functions behave the way they do. In our example, the function in question is y = \( \sqrt{x} + 4 \). A square root function typically takes the form f(x) = \( \sqrt{x} \), and it denotes the principal square root of x. This means we're looking for a non-negative number that, when multiplied by itself, gives x.

The defining characteristic of a square root function is that it is not defined for negative numbers under the square root symbol, or 'radical.' Since we cannot take the square root of a negative number and obtain a real number result, the domain of \( \sqrt{x} \) is restricted to x \geq 0. This limitation on the domain directly impacts the shape and position of its graph on a coordinate plane.

Sketching Graphs of Functions

Sketching the graph of a function like y = \( \sqrt{x} + 4 \) involves comprehending not just the domain and range, but also the general shape or curve represented by the function's equation. To start, we should always plot any known points; for the square root function, the origin point (0, 4) is such a point, as the range informs us that the function's value cannot go below 4.

Visualizing how the function behaves as x increases helps us further. The graph of a pure square root function looks like the right half of a parabola lying on its side. However, since y is a square root function shifted upwards by 4, its graph will start 4 units above the origin on the y-axis. With these steps in mind, we can sketch a rising curve that starts at (0, 4) and extends onward within the first quadrant of the graph, asymptotically approaching but never touching the x-axis.

Mathematical Intervals

In mathematics, intervals are used to describe a set of numbers between two endpoints. They can be 'closed,' 'open,' or a combination of both, indicating whether the endpoints are included in the set. The domain and range of a function are often expressed using interval notation.

For the given function y = \( \sqrt{x} + 4 \), the domain is represented as [0, +\infty), where the square bracket [ means the number 0 is included (also described as 'closed' at 0), and the parenthesis ( indicates that infinity is not a number that can be reached but rather a concept that depicts an unbounded limit (thus 'open' at infinity). Similarly, the range is [4, +\infty), closed at 4 because the function can actually equal 4, and open at infinity, suggesting the function's values can grow without bound.