Question

Find (a) The domain. (b) The range. $$ f(x)=\frac{1}{x+1}+3 $$

Short answer

Answer: (a) The domain is $$(-\infty, -1) \cup (-1, \infty)$$. (b) The range is $$(3, \infty)$$.

Step-by-step solution

Identify any restrictions for the domain

In the given function, the denominator is x+1, so the only restriction we need to be concerned with is where this denominator is equal to zero. We find this value by solving: $$ x+1=0 $$ Solving for x, we have: $$ x=-1 $$ The denominator would be zero when x=-1, so we need to exclude this value from the domain.

Determine the domain

Since the only restriction on x is that x can't be -1, the domain of the function will be all real numbers except x = -1. In interval notation, the domain can be written as: $$ (-\infty, -1) \cup (-1, \infty) $$

Analyze the behavior of the function

To find the range, we first need to analyze the function's behavior. As x approaches infinity, the term $$\frac{1}{x+1}$$ approaches 0, which means the function approaches 3. Similarly, as x approaches negative infinity, the term $$\frac{1}{x+1}$$ approaches 0, and the function approaches 3. When x is not equal to -1, the term $$\frac{1}{x+1}$$ will always be greater than 0 as the denominator cannot be 0, so the function will always be greater than 3.

Determine the range

Since the function is always greater than 3 but approaches 3 from both directions, the range will be all real numbers greater than 3. In interval notation, the range can be written as: $$ (3, \infty) $$ To summarize: (a) The domain is $$(-\infty, -1) \cup (-1, \infty)$$. (b) The range is $$(3, \infty)$$.

Key concepts

Function Notation

Function notation is a way of representing functions in mathematics, offering a more structured method of denoting the relationship between variables. When you see something like \(f(x)\), this is function notation. Here, \(f\) represents the name of the function, and \(x\) is the variable inside it. This tells us that \(f\) is a function in terms of \(x\).

This notation is useful because it helps specify inputs and outputs clearly. For example, for the function \(f(x) = \frac{1}{x+1} + 3\), \(x\) is the input, and the output \(f(x)\) depends on this value. Understanding this setup is crucial because it helps in identifying how changing \(x\) affects the output.

• \(x\) is the independent variable.
• \(f(x)\) is the dependent variable, as it depends on \(x\).

Function notation makes it easier to analyze functions' behavior, allowing us to substitute values into \(x\) to find specific outputs.

In our example, when determining the domain, we could quickly see that for \(x = -1\), \(f(x)\) would be undefined, which helps in finding restrictions.

Denominator

In a rational function like \(f(x) = \frac{1}{x+1} + 3\), the denominator is the expression located at the bottom of the fraction. It's essential to pay close attention to the denominator because it can't be zero — otherwise, the function becomes undefined.

To determine where our function is undefined, set the denominator equal to zero and solve for \(x\). In our function, this equation is \(x + 1 = 0\). Solving it gives us \(x = -1\), which is our restriction. We exclude this value from our domain of the function. This is because dividing by zero isn’t allowed, leading to an undefined expression.

Remember:

• Always solve \(\text{denominator} = 0\) to find restrictions.
• Exclude these points from the domain.

This care with denominators helps avoid undefined points and ensures we represent the function accurately.

Interval Notation

Interval notation is a shorthand used to describe the domain and range of functions clearly and efficiently. It uses parentheses and brackets to denote open and closed intervals.

In our exercise, the domain and range are described using interval notation:

• The domain is \((-\infty, -1) \cup (-1, \infty)\), indicating all real numbers except \(x = -1\).
• The "\((-\infty, -1)\)" part shows that numbers approach but do not include \(-1\).
• The range is \((3, \infty)\), suggesting the function's output is greater than 3.

Understanding the symbols:

• "(") and ")" denote that the endpoint is not included (open interval).
• "[" and "]" indicate that the endpoint is included (closed interval).
• "\(\cup\)" is the union symbol, used to join separate intervals.

Interval notation simplifies the process of defining sets of numbers, offering a clear method to express both the scope and limitations of functions.