Problem 7 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. [Hint: Use a graphing calculator. You may have to ignore some false lines on the graph. Graphing in "dot mode" will also eliminate false lines. $$ f(x)=\frac{12}{x(x+4)} ; \text { find } f(2) $$

Step-by-Step Solution

Answer

Evaluate: 1; Domain: \((-\infty, -4) \cup (-4, 0) \cup (0, \infty)\); Range: \((-\infty, \infty)\).

Step 1: Evaluate the Expression

To evaluate the expression at \( x = 2 \), substitute 2 into the function \( f(x) = \frac{12}{x(x+4)} \). This gives \( f(2) = \frac{12}{2(2+4)} = \frac{12}{12} = 1 \). Therefore, \( f(2) = 1 \).

Step 2: Find the Domain of the Function

The domain of the function \( f(x) = \frac{12}{x(x+4)} \) is the set of all real numbers except where the denominator is zero. Set the denominator \( x(x+4) = 0 \) and solve for \( x \). This gives \( x = 0 \) and \( x = -4 \). Thus, the domain is all real numbers except \( x = 0 \) and \( x = -4 \), written as \( x eq 0 \) and \( x eq -4 \), or in interval notation: \( (-\infty, -4) \cup (-4, 0) \cup (0, \infty) \).

Step 3: Find the Range of the Function

To find the range of \( f(x) = \frac{12}{x(x+4)} \), note that the function is a rational function and its range is affected by vertical asymptotes at \( x = 0 \) and \( x = -4 \). Using graphing tools or by analysis, recognize that the function values can approach any value close to \( \pm\infty \) as \( x \) approaches the asymptotes or goes to infinity. Therefore, the range is all real numbers, \( (-\infty, \infty) \).

Key Concepts

Function evaluation

When it comes to function evaluation, we're simply finding out what value a function outputs when we plug a specific number into it. For instance, let's consider a function, say the rational function given by \[ f(x) = \frac{12}{x(x+4)} \].To evaluate this particular expression, we're tasked with replacing the variable "x" with the number we're interested in—here, it's 2. This means doing the math for \[ f(2) = \frac{12}{2(2+4)} = \frac{12}{12} \].After simplifying, we find that \( f(2) = 1 \).

• So, function evaluation is all about replacement and simplification.
• It's important to perform each arithmetic operation carefully to avoid mistakes.
• Every step needs attention to detail to ensure accuracy, especially in complex functions.

Domain of a function

The domain of a function refers to all the possible input values "x" for which the function is defined. In other words, it's answering the question, "What values can I plug into this function without causing any trouble like division by zero?"
For the function \[ f(x) = \frac{12}{x(x+4)} \], the denominator cannot be zero since division by zero is undefined. So, we set the denominator equal to zero and solve: \[ x(x+4) = 0 \] This results in \( x = 0 \) or \( x = -4 \).
Thus, the domain excludes these numbers and is expressed in interval notation as: \( (-\infty, -4) \cup (-4, 0) \cup (0, \infty) \).

• The domain is crucial because it specifies the input limitations of the function.
• Always check for values that could make the denominator zero in rational functions.
• Use set notation or interval notation to express domains clearly.

Range of a function

The range of a function consists of all possible output values it can produce. For rational functions like \[ f(x) = \frac{12}{x(x+4)} \], this can be a bit tricky to determine just by looking at the equation. Often, graphs provide excellent visual support.
For this function, there are vertical asymptotes at \( x = 0 \) and \( x = -4 \). An asymptote is a line that the graph of the function approaches infinitely close but never actually reaches.
Using a Graph:- Graphing tools can help visualize what's going on.- The function values get extremely large, approaching \( \pm\infty \) as \( x \) goes near the asymptotes or extends to \( \pm\infty \).- Considering this, the range is expressed as \( (-\infty, \infty) \), which means the function can achieve any real number.

• The range can often depend on the horizontal and vertical behavior of the graph of the function.
• Vertical asymptotes influence the range significantly in rational functions.
• Graphing tools can be extremely helpful for understanding the range.