Question

Find (a) The domain. (b) The range. $$ y=\sqrt{x}+1 $$

Short answer

Answer: (a) Domain: \(x \geq 0\); (b) Range: \(y \geq 1\)

Step-by-step solution

Analyze the function

The given function is of the form \(y = \sqrt{x} + 1\). This function is a transformation of the basic square root function \(y = \sqrt{x}\), where 1 is added to the square root part.

Determine the domain of the original square root function

Recall that the square root of a number can only be found if the number inside the square root is greater than or equal to 0. Therefore, for the function \(y = \sqrt{x}\), the domain is all non-negative real numbers. In inequality form, the domain can be represented as: \[x \geq 0\]

Determine the domain of the transformed function

Adding a constant term to the square root function does not affect its domain. The constant only affects the y-values, not the x-values. Therefore, the domain of the function \(y = \sqrt{x} + 1\) is the same as that of the original square root function, which is all non-negative real numbers or in inequality form: \[x \geq 0\]

Determine the range of the original square root function

For the original square root function \(y = \sqrt{x}\), the output value, y, will always be greater than or equal to 0 since the square root of a non-negative value is non-negative. In inequality form, the range can be represented as: \[y \geq 0\]

Determine the range of the transformed function

In the transformed function \(y = \sqrt{x} + 1\), the constant term, 1, is added to the square root part. Thus, the minimum value of y is now 1. For a range, it can be described in inequality form as: \[y \geq 1\]

Final Answer

Based on the steps above, the domain and range of the given function are as follows: (a) Domain: \(x \geq 0\) (b) Range: \(y \geq 1\)

Key concepts

Square Root Function

The square root function is a basic and essential mathematical function often represented by the formula \(y = \sqrt{x}\). The primary goal of this function is to find the non-negative root of a given non-negative number. In simpler terms, it's the positive number that, when squared, gives you the original value under the square root.

• For example, when \(x = 4\), \(y = \sqrt{4} = 2\).
• It's important to note that the square root function is only defined for values of \(x\) that are zero or greater. This is because the square root of a negative number isn't a real number.
• Therefore, the domain of the square root function is always \(x \geq 0\).

Next, you might wonder about its range. For the square root function \(y = \sqrt{x}\), no negative outputs are possible since we start with non-negative inputs. This means the range is \(y \geq 0\). Understanding this lays a strong foundation as we transform these functions in more complex ways.

Transformation of Functions

Transformations allow us to change basic functions into newer, often more complex, variations. When we look at \(y = \sqrt{x} + 1\), we're applying a transformation to the square root function. But how?

• Here, we added a constant 1 to the basic function \(y = \sqrt{x}\).
• This operation is known as vertical translation or shift.
• Adding 1 moves the entire graph of \(y = \sqrt{x}\) up by 1 unit on the y-axis.

What's important to remember is that such vertical shifts do not affect the domain. The domain remains the same, \(x \geq 0\), since the condition for \(x\) hasn't changed. Transformation shifts the output (y-values), resulting in our new function, \(y = \sqrt{x} + 1\), having its range adjusted to \(y \geq 1\). Each transformation adjusts how the function behaves, whether it's stretching, compressing, or shifting.

Inequality Representation

Inequality representation is a critical part of understanding and expressing the domain and range of functions. When working with functions like \(y = \sqrt{x} + 1\), representing conditions such as domain and range effectively is essential for clarity.

• The domain, here represented as \(x \geq 0\), signifies that all values of \(x\) are zero or positive. It's due to the fact that you can't take the square root of a negative number in real numbers.
• The range, written as \(y \geq 1\), indicates that the lowest possible output of our transformed function is 1, owing to the addition of the constant 1 to the square root value.

Such inequalities help in clearly specifying which input values (domain) result in valid output values (range). Understanding and using these inequality forms is crucial when discussing or solving problems involving functions, as they precisely describe conditions for elements of the function to be defined.