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Chapter 3: Problem 16

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\frac{\sqrt{x+4}}{x-4} $$

Short Answer

Expert verified
Domain: \([-4, 4) \cup (4, \infty)\)

Step by step solution

01

Recognize Function Type and Constraints

The function \( f(x) = \frac{\sqrt{x+4}}{x-4} \) contains a square root in the numerator and a linear expression in the denominator. To find the domain, we need to ensure that all parts of the function are defined. This means:1. The expression \( x+4 \) under the square root must be non-negative.2. The denominator \( x-4 \) cannot equal zero.
02

Determine Square Root Constraint

For the square root expression \( \sqrt{x+4} \) to be defined, the inside, \( x+4 \), must be greater than or equal to zero. Solve the inequality:\[ x+4 \geq 0 \]Subtract 4 from both sides:\[ x \geq -4 \]This means \( x \) must be greater than or equal to \(-4\) for the square root to be valid.
03

Determine Denominator Constraint

The denominator must not be zero to ensure the function is defined at all points:\[ x-4 eq 0 \]Solve this by adding 4 to both sides:\[ x eq 4 \]This means \( x \) cannot be 4.
04

Combine Constraints

Combine the results from Steps 2 and 3 to find the domain of the function. The domain is where both conditions are satisfied:- \( x \geq -4 \)- \( x eq 4 \)So the domain of \( f(x) \) in interval notation is \([-4, 4) \cup (4, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function is often represented as \( \sqrt{x} \) in algebra. It involves taking the square root of a number or expression. For the square root function to be defined, the expression inside the square root must be non-negative. This is because square roots of negative numbers are not real. Here are some key points:
  • The expression inside the square root (radicand) must be zero or positive.
  • Determine the constraints on the variable by solving the inequality for the radicand.
In the function \( f(x) = \frac{\sqrt{x+4}}{x-4} \), the square root portion is \( \sqrt{x+4} \). Therefore, we solve \( x+4 \geq 0 \), leading to \( x \geq -4 \). This ensures the radicand is non-negative, so the square root is valid.
Interval Notation
Interval notation is a mathematical way to represent a range of values for which a function is defined. It's a concise method to describe inequalities related to functions. Here are its components:
  • Brackets \( [ ] \) indicate the endpoints are included in the interval.
  • Parentheses \( ( ) \) imply the endpoints are not included.
For the function given, we sought the domain through constraints. We found that \( x \geq -4 \) and \( x eq 4 \). Therefore, the domain in interval notation is expressed as \([-4, 4) \cup (4, \infty)\). This indicates that \( x \) includes values from \(-4\) to just below \(4\), and values above \(4\) extending to infinity, aligning all conditions.
Rational Function
A rational function is expressed as the quotient of two polynomials. The function \( f(x) = \frac{\sqrt{x+4}}{x-4} \) is also a rational function. When dealing with rational functions:
  • Identify any restrictions where the denominator equals zero, as these points are undefined.
  • Analyze the numerator for additional constraints (like square roots).
Here, the denominator of \( f(x) \) is \( x-4 \). This implies \( x eq 4 \), as the function would be undefined at this point. Additionally, combining this with the requirement for the square root, we obtain the domain \([-4, 4) \cup (4, \infty)\). This takes into account both the numerator and the denominator's restrictions.

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Most popular questions from this chapter

For the following exercises, use the values listed in Table 6 to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\\ \hline \end{array} $$ Find \(f(1)\).

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions. $$ f(x)=-3 x+5 \text { and } g(x)=\frac{x-5}{-3} $$

For the following exercises, find a domain on which each function \(f\) is one- to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain. $$ f(x)=(x-6)^{2} $$

Are one-to-one functions either always increasing or always decreasing? Why or why not?

Use each pair of functions to find \(f(g(0))\) and \(g(f(0))\). $$ f(x)=4 x+8, g(x)=7-x^{2} $$
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