Problem 77 Determine the domain of each fun... [FREE SOLUTION]

Chapter 12: Problem 77

Determine the domain of each function. $$k(x)=\frac{1}{x^{2}+11 x+24}$$

Short Answer

Expert verified

The domain of the function $k(x) = \frac{1}{x^2 + 11x + 24}$ is all real numbers except for x = -3 and x = -8, which can be written in interval notation as: $(-\infty, -8) \cup (-8, -3) \cup (-3, \infty)$.

Step by step solution

Identify the denominator of the function

We are given the function $k(x) = \frac{1}{x^2 + 11x + 24}$. The denominator of this function is $x^2 + 11x + 24$.

Find the values of x that make the denominator equal to zero

To find the values of $x$ that make the denominator equal to zero, we need to solve the following equation: $x^2 + 11x + 24 = 0$. To solve this quadratic equation, we can either use factoring or the quadratic formula. Let's try factoring first: $(x^2 + 11x + 24) = (x+3)(x+8)$ Now, set each factor equal to zero: $x + 3 = 0 => x = -3$ $x + 8 = 0 => x = -8$ So, there are two values of $x$ that make the denominator equal to zero: $x = -3$ and $x = -8$.

Determine the domain of the function

Since the denominator cannot be equal to zero, we have to exclude the values of $x$ that we found in step 2 from the domain of the function. Therefore, the domain of the function $k(x)$ is all real numbers except for $x = -3$ and $x = -8$. We can write this in interval notation as: Domain: $(-\infty, -8) \cup (-8, -3) \cup (-3, \infty)$

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Key Concepts

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

Quadratic Equation

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a eq 0$. This type of equation is fundamental in algebra and describes a parabolic relationship within a function.

It features the square of the unknown variable $x$, hence "quadratic" from the Latin word "quadratus," meaning square.
The general solutions for a quadratic equation can be found using the quadratic formula, factoring, or completing the square.

In our exercise, the quadratic equation given in the denominator of the function is $x^2 + 11x + 24$. Solving this equation is crucial as it determines the x-values where the function is undefined.

Factoring

Factoring involves breaking down an expression into simpler "factors" or components that, when multiplied together, give back the original expression. It is a useful technique for solving quadratic equations.

This involves rewriting the quadratic equation in the form $(x+p)(x+q)$, where $p$ and $q$ are numbers such that $p+q = b$ and $pq = c$ in the standard form $ax^2 + bx + c$.
When we factor $x^2 + 11x + 24$, it becomes $(x+3)(x+8)$.

By setting each factor equal to zero, we solve for the roots of the equation, which are $x = -3$ and $x = -8$. These roots tell us where the denominator equals zero and are crucial in determining the function's domain.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is very handy when expressing the domain or range of a function.

Closed intervals [a, b] include the endpoints $a$ and $b$, meaning every number between $a$ and $b$ is included.
Open intervals $(a, b)$ do not include the endpoints a and b, thus including only the numbers strictly between $a$ and $b$.
Union of intervals, denoted by $\cup$, is used to combine disjoint intervals necessary to express a complete domain.

For example, for the function $k(x)$, the domain is all real numbers except where the denominator is zero. The domain can therefore be written in interval notation as $(-\infty, -8)\cup(-8,-3)\cup(-3,\infty)$. This clearly shows that the function includes all real numbers except $x = -8$ and $x = -3$.

Real Numbers

Real numbers include all the numbers on the number line. This set encompasses both rational numbers (such as 2, 1/2, 0.75) and irrational numbers (such as $\pi$, $\sqrt{2}$). Real numbers are essential when discussing the domain of functions because they describe all possible values that $x$ can take under typical circumstances.

The domain of a typical function, unless specified, is usually the real numbers, excluding any restrictions like division by zero or the square roots of negative numbers.

In the context of our function $k(x) = \frac{1}{x^2 + 11x + 24}$, we state that the domain is "all real numbers" except for the values $x = -3$ and $x = -8$ because these values would make the function's denominator zero, making the function undefined.

91影视

Determine the domain of each function. $$k(x)=\frac{1}{x^{2}+11 x+24}$$

Short Answer

Step by step solution

Identify the denominator of the function

Find the values of x that make the denominator equal to zero

Determine the domain of the function

Key Concepts

Quadratic Equation

Factoring

Interval Notation

Real Numbers

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