Problem 13 - Algebra 1

Question

Find the domain and the range of the function. $$y=\sqrt{x+5}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\sqrt{x+5}\) is \([-5,\infty)\) and the range is \([0,\infty)\).

Step 1: Find the Domain

Determine what x-values will make the function work. This requires solving the inequality for x where the expression under the square root (i.e., \(x+5\)) is greater than or equal to 0. Solve for x and we find \(x\geq-5\), so the domain is \([-5,\infty)\).

Step 2: Find the Range

Think about what y-values are possible outputs of the function. The square root function can output all non-negative numbers, so the range is \([0,\infty)\).

Key Concepts

Square Root Function

A square root function is a type of function that involves the square root of a variable. The general form is \( y = \sqrt{x} \). In the given exercise, it's slightly different since we have \( y = \sqrt{x+5} \), meaning we're taking the square root of \( x \) plus 5. This function looks like a stretched curve starting from a specific point on the coordinate plane.
This point is determined based on what makes the expression under the square root non-negative. For any square root function, like \( \sqrt{x+5} \), the expression inside the root must be zero or more.

• For example, \( x + 5 \geq 0 \) ensures that the square root is defined.
• This is typically the first step in analyzing these functions: finding where they "begin" on a graph.

It's vital to remember that square root functions are only defined for those indoor values that make the inside non-negative. This characteristic directly leads us to explore their domain and range.

Inequalities

Understanding inequalities is crucial when determining the domain of square root functions. An inequality shows how one expression is related to another through signs like greater than (\(>\)), less than (\(<\)), or equal to (\(\leq, \geq\)).
When you deal with square root functions like \(y = \sqrt{x+5}\), you ensure that the expression inside the square root is never negative.

• This requires setting up an inequality: \(x + 5 \geq 0\).
• Solving this, you subtract 5 from both sides and find \(x \geq -5\).

This action shapes the domain of the function, which is all the \(x\) values that satisfy the inequality. Inequalities help us identify possible input values, ensuring that our mathematical operations are valid.
Being comfortable with setting up and solving inequalities is an asset when working with many mathematical functions.

Function Outputs

The outputs of a function are the possible values that the function can produce, commonly referred to as the range. For the square root function \( y = \sqrt{x+5} \), we examine potential output values based on the inputs that fall within the domain.
Here's how you determine the range:

• From your work determining the domain, you know \( x \geq -5 \).
• Substituting the lowest \(x\), \(-5\), into \(\sqrt{x + 5}\), gives \(y = 0\).

As \(x\) increases toward infinity, so does \(y\). Thus, the smallest output value is 0, but there's no upper limit.
This results in the function's range being \([0, \infty)\). Thus, function outputs in square root functions start at 0 and increase without bound as the input \(x\) increases.