Problem 38 - Algebra A Combined Function

Question

If \(f(x)=\frac{x^{2}+5}{x}\) and \(g(x)=\frac{x^{2}+2 x}{x+3},\) find each function value. $$ \text { Find the domain of } g(x) \text { . } $$

Step-by-Step Solution

Answer

The domain of \(g(x)\) is \((-\infty, -3) \cup (-3, \infty)\).

Step 1: Determine the domain of g(x)

The domain of a function is the set of all possible input values (x-values) that the function can accept without causing division by zero or any other undefined operation. For the function \(g(x)=\frac{x^2+2x}{x+3}\), we need to find the values of \(x\) that make the denominator zero, as these values are not allowed in the domain. Here, the denominator is \(x+3\). Setting \(x+3=0\) gives \(x=-3\). Thus, the domain includes all real numbers except \(x=-3\).

Step 2: Express the domain in interval notation

After determining the values that are excluded from the domain, we can express the domain using interval notation. Since \(x=-3\) is not included, the domain of \(g(x)\) can be written in interval notation as \((-\infty, -3) \cup (-3, \infty)\). This denotes all real numbers except \(x=-3\).

Key Concepts

Domain of a Function

The domain of a function is a fundamental concept in mathematics. It's essentially the set of all possible input values or "x-values" that a function can accept. For any given function, you want to make sure that plugging these x-values into the function doesn't lead to undefined results—things like division by zero or the square root of a negative number. To find the domain for a given function like \(g(x)=\frac{x^2+2x}{x+3}\), you need to first focus on the denominator. Since division by zero is undefined, any number that makes the denominator zero needs to be avoided. So, in this case, you solve \(x+3=0\), which gives \(x=-3\). This means \(x=-3\) is not in the domain because it would make the denominator zero, leading to an undefined value. Thus, the domain of \(g(x)\) is all real numbers except \(x=-3\). You can write this more formally using interval notation, which we'll discuss in the next section.

Interval Notation

Interval notation is a convenient way to describe the domain of a function. It succinctly tells you which values of x are included or excluded. Instead of saying "all real numbers except x=-3," interval notation uses parentheses and brackets to clearly convey this information.- Parentheses \(()\) are used to indicate that a number is not included in the domain. - Brackets \([])\) are used when a number is included. - The symbol \((\cup)\) is used to denote a union, meaning "all numbers in either set."When we express the domain of \(g(x)=\frac{x^2+2x}{x+3}\) in interval notation, it becomes \((-\infty, -3) \cup (-3, \infty)\). This tells us:

• Consider all the numbers less than \(-3\).
• And all the numbers greater than \(-3\).

There is a gap or "hole" in the domain at \(-3\), where the function is undefined because it would cause division by zero. This clear and structured way of stating domains for functions is especially useful in calculus and higher-level mathematics.

Division by Zero

Division by zero is a common mistake to watch out for when working with functions. Why does it matter so much? Well, mathematically speaking, division by zero is undefined because it doesn't produce a meaningful result. When you divide a number by zero, it leaves you in a kind of mathematical limbo where the laws of arithmetic just don't apply. For example, in the function \(g(x)=\frac{x^2+2x}{x+3}\), if \(x=-3\), the denominator becomes zero: \((x+3 = 0)\). If you were to substitute \((-3)\) into \(g(x)\), the result would be an undefined expression, as dividing by zero has no consistent value. To avoid this, it's crucial to recognize and eliminate these possibilities when determining a function's domain. By excluding these values, such as \(x=-3\) in this scenario, we ensure the function remains valid and safe to use across its domain.