Problem 25
Question
For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{x^{2}-9 x}{x^{2}-81}\)
Step-by-Step Solution
Verified Answer
The domain of \( f(x) \) is \((-\infty, -9) \cup (-9, 9) \cup (9, \infty)\).
1Step 1: Understanding the Function
The function given is a rational function \( f(x) = \frac{x^2 - 9x}{x^2 - 81} \). For rational functions, the domain consists of all real numbers except those where the denominator is zero.
2Step 2: Finding Zeroes of the Denominator
Set the denominator equal to zero and solve for \( x \): \( x^2 - 81 = 0 \). This can be written as \( (x-9)(x+9) = 0 \). Thus, \( x = 9 \) and \( x = -9 \) are the zeros of the denominator.
3Step 3: Excluding Zeroes from the Domain
The values \( x = 9 \) and \( x = -9 \) make the denominator zero, so we must exclude them from the domain of \( f(x) \).
4Step 4: Writing the Domain in Interval Notation
To express the domain in interval notation, exclude \( x = 9 \) and \( x = -9 \) from the real numbers. Therefore, the domain of \( f(x) \) is \((-\infty, -9) \cup (-9, 9) \cup (9, \infty)\).
Key Concepts
Understanding Interval NotationZeroes of the DenominatorUnderstanding Rational FunctionsDomain Exclusion in Rational Functions
Understanding Interval Notation
When we talk about interval notation in mathematics, we mean a way to describe subsets of the real numbers. This is especially useful when we're defining the domain of a function. The domain comprises all possible input values for which the function is defined.
To use interval notation, we look at the numbers that our function can legally "eat," considering any restrictions like division by zero. For such issues, parentheses are used to show that a particular number is not included, while square brackets indicate that a number is included.
For example, in our specific problem, because the values of x equal to 9 and -9 need to be excluded (since they make the function undefined), the domain in interval notation is presented as
To use interval notation, we look at the numbers that our function can legally "eat," considering any restrictions like division by zero. For such issues, parentheses are used to show that a particular number is not included, while square brackets indicate that a number is included.
For example, in our specific problem, because the values of x equal to 9 and -9 need to be excluded (since they make the function undefined), the domain in interval notation is presented as
-
(-∞, -9)
- This includes all numbers less than -9.
- Exclude exactly the points -9 and 9 with (-9, 9).
-
(9, ∞)
- This captures all numbers greater than 9.
Zeroes of the Denominator
Identifying the "zeroes of the denominator" is crucial when working with rational functions. These zeroes are numbers that make the denominator equal to zero, creating division by zero, which is undefined in mathematics.
In the exercise, the denominator of our function is given as \(x^2 - 81\). To find these problematic values, set this equal to zero: \(x^2 - 81 = 0\).
Solving this equation, you factor it to \((x-9)(x+9) = 0\). Therefore, the zeroes are \(x = 9\) and \(x = -9\).
These specific x-values are where our function will misbehave because it leads to division by zero, thus needing exclusion from the domain for our function to be defined and meaningful.
In the exercise, the denominator of our function is given as \(x^2 - 81\). To find these problematic values, set this equal to zero: \(x^2 - 81 = 0\).
Solving this equation, you factor it to \((x-9)(x+9) = 0\). Therefore, the zeroes are \(x = 9\) and \(x = -9\).
These specific x-values are where our function will misbehave because it leads to division by zero, thus needing exclusion from the domain for our function to be defined and meaningful.
Understanding Rational Functions
Rational functions are ratios of two polynomials, where you have one polynomial divided by another. A typical representation of a rational function looks like \(f(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) isn't zero.
These functions are defined everywhere except where the denominator equals zero. That is because dividing by zero isn't possible in the framework of real numbers.
In our exercise, the function \(f(x) = \frac{x^2 - 9x}{x^2 - 81}\) is a classic rational function. It becomes undefined specifically at the zeroes of the denominator, showing why identifying those zeroes, like we did, is important when considering the domain of a rational function.
These functions are defined everywhere except where the denominator equals zero. That is because dividing by zero isn't possible in the framework of real numbers.
In our exercise, the function \(f(x) = \frac{x^2 - 9x}{x^2 - 81}\) is a classic rational function. It becomes undefined specifically at the zeroes of the denominator, showing why identifying those zeroes, like we did, is important when considering the domain of a rational function.
Domain Exclusion in Rational Functions
When determining the domain of a rational function, recognizing domain exclusions is vital. This means figuring out which real number inputs must be left out to ensure the function remains defined. Generally, these are values where the denominator becomes zero.
As we solved, for \(x^2 - 81\), the zeroes \(x = 9\) and \(x = -9\) create undefined moments since they would result in division by zero.
To accurately describe where the function is defined, we exclude these values from the domain using interval notation. In our problem, excluding these values results in
As we solved, for \(x^2 - 81\), the zeroes \(x = 9\) and \(x = -9\) create undefined moments since they would result in division by zero.
To accurately describe where the function is defined, we exclude these values from the domain using interval notation. In our problem, excluding these values results in
- (-∞, -9)
- Which shows all values below -9 are fine.
- (-9, 9)
- (9, ∞)
- Reflecting all values above 9 are also acceptable.
Other exercises in this chapter
Problem 25
For the following exercises, graph the given functions by hand. \(f(x)=2|x+3|+1\)
View solution Problem 25
For \(f(x)=\frac{1}{x}\) and \(g(x)=\sqrt{x-1},\) write the domain of \((f \circ g)(x)\) in interval notation.
View solution Problem 25
For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y^{2}=x^{2}\)
View solution Problem 26
For the following exercises, graph the given functions by hand. \(f(x)=3|x-2|+3\)
View solution