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Rational functions might sound complicated, but they're pretty straightforward once you break them down. A rational function is simply a ratio of two polynomials. A common example looks like this: \( h(x) = \frac{x}{x+1} \).
The key thing to remember is that the function is undefined wherever the denominator equals zero, because you can't divide by zero.
To determine where the function is undefined, you'll need to find the values of \(x\) that make the denominator equal to zero.
For the function \( h(x) = \frac{x}{x+1} \), you'd set the denominator \( x+1 \) to zero and solve for \( x \).
This gives you: \( x + 1 = 0 \)
\( x = -1 \).
So, the function is undefined at \(x = -1\). This step is crucial for finding the domain.

Finding the domain of a function means identifying all possible values of \( x \) that you can plug into the function without causing any undefined situations. For rational functions, you'll often look for values that make the denominator zero, because these are the values where the function is undefined.
Here's a quick walkthrough:

• First, identify the denominator in the function. For \( h(x) = \frac{x}{x+1} \), the denominator is \( x + 1 \).
• Next, find what values of \( x \) make this denominator zero by solving the equation: \( x + 1 = 0 \).
• Then, solve for \( x \): \( x = -1 \).
• State the domain. The function will be defined for all other real numbers except where the denominator is zero. So, the domain of \( h(x) \) is all real numbers except \( x = -1 \).

This step-by-step approach helps ensure nothing is overlooked.

Interval notation is a simple, compact way to describe a range of values on a number line. It's particularly useful for expressing the domain of a function. Here's how it works:
When you need to exclude a certain value from the domain, like \( x = -1 \) in the function \( h(x) = \frac{x}{x+1} \), interval notation can efficiently show this exclusion.
For the domain of \( h(x) \), which is all real numbers except -1, you would write it as:

• \( (-\infty, -1) \cup (-1, \infty) \)

The symbols \( (-\infty, -1) \) mean all real numbers from negative infinity up to but not including -1, and \( (-1, \infty) \) means all real numbers greater than -1 up to positive infinity. The union symbol \( \cup \) is used to combine these two intervals, showing all values except -1.

This makes it very clear and easy to understand, ensuring you don't accidentally include values where the function is undefined.