Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the domain of \(f(x)=\frac{\sqrt{x+2}}{x-4}\)

Short answer

The domain of \(f(x)=\frac{\sqrt{x+2}}{x-4}\) is \[[-2, 4)\cup(4, \infty)\].

Step-by-step solution

Identify restrictions from the numerator

The numerator is \(\textbackslash sqrt{x+2}\). A square root is only defined when its operand is non-negative. Therefore, we need to solve the inequality: \({x+2\geq0}\) or \({x\geq-2}\).

Identify restrictions from the denominator

The denominator is \( {x-4}\). A fraction is undefined when its denominator is zero. Therefore, we need to solve the equation: \({x-4\eq0}\) or \({x\eq4}\).

Combine restrictions

Combine the constraints from the previous steps. So, \({x\geq-2}\) and \({x\eq4}\). This means that \(x\) can be any value greater than or equal to \(-2\), except \(x=4\).

Express the domain in interval notation

The domain can now be expressed in interval notation considering all restrictions: \[[-2, 4)\cup(4, \infty)\].

Key concepts

Domain Restrictions

When we talk about the domain of a function, we mean all the possible input values (usually represented as x) for which the function is well-defined. In other words, it's the set of all x-values that you can plug into a function without running into any problems like division by zero or taking the square root of a negative number.

For example, given the function:\[f(x)=\frac{\sqrt{x+2}}{x-4}\]here are the domain restrictions you need to consider:

• Square Root Constraints: The square root function, indicated by \sqrt{x+2}, is only defined for non-negative numbers. This means that the expression inside the square root must be zero or positive. Therefore, we set up the inequality:
  \[{x+2\geq0}\]
  which simplifies to
  \[{x\geq-2}\].
• Denominator Constraints: The denominator (x-4) must never be zero because division by zero is undefined. We find where the denominator equals zero by solving the equation:
  \[{x-4=0}\]
  which means:
  \[{x=4}\]. This tells us that x cannot be 4.

Interval Notation

Once you've identified the domain restrictions, the next step is to express the domain in interval notation. Interval notation is a concise way to describe a range of values along a number line.

Here, we're interested in two things:

• Values of x that are greater than or equal to \[{ -2 }\].
• Excluding the value \[{ 4 }\] from the domain.

So, how do we write this in interval notation?

We split this into two intervals to avoid including 4:
1. From \[{ -2 }\] up to, but not including, 4:
\[[-2, 4)\]
2. From just after 4 to infinity:
\[(4, \infty )\]
Combining these, the domain of the function in interval notation becomes:
\[[-2, 4)\cup(4, \infty ) \]

Square Root Function

The square root function, represented as \sqrt{x}, is one of the most common types of functions you’ll encounter. This function is defined only for non-negative values of x. This means that the expression under the square root must be zero or positive to be valid.

For example, in our function:\[\frac{\sqrt{x+2}}{x-4}\]
We need to make sure that \sqrt{x+2} has real, non-negative values. This results in the inequality:
\[{x+2\geq0} \]
or simply:
\[{x\geq-2} \]
When solving problems involving square root functions, always remember to check and include the restriction where the operand of the square root must be greater than or equal to zero. This ensures the values stay within the realm of real numbers.