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Chapter 12: Problem 79

Determine the domain of each function. $$r(c)=\frac{c+3}{c^{2}-5 c-36}$$

Short Answer

Expert verified
The domain of the function \(r(c)=\frac{c+3}{c^{2}-5c-36}\) is \(\{c | c \in \mathbb{R}, c \neq -4, c \neq 9\}\), which means it includes all real numbers except for \(c = -4\) and \(c = 9\).

Step by step solution

01

Factor the denominator

The first step is to factor the denominator, which is a quadratic expression. Let's factor \(c^2 - 5c - 36\): \[(c - 9)(c + 4)\]
02

Solve for \(c\)

Now we'll set each of the factors equal to zero and solve for \(c\): c - 9 = 0 and c + 4 = 0 c = 9 and c = -4
03

Determine the domain

Since the function is undefined when the denominator is zero, we'll exclude these values from the domain. The domain of the function is given by: \[\{c | c \in \mathbb{R}, c \neq -4, c \neq 9\}\] So, the domain of the function \(r(c)=\frac{c+3}{c^{2}-5c-36}\) is all real numbers except for \(c = -4\) and \(c = 9\).

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Key Concepts

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

Polynomials
When we talk about polynomials, we're discussing expressions made up of variables and coefficients, all tied together using operations like addition, subtraction, and multiplication. A polynomial can be as simple as a single number (called a constant) or involve multiple variables raised to different powers.
Here are a few key ideas about polynomials:
  • Each part of a polynomial separated by a plus or minus sign is called a "term."
  • The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial \(2x^3 + 3x^2 - x + 4\), the degree is 3 because the highest power of \(x\) is 3.
  • Polynomials can have several variables, like \(x\) and \(y\), and are then called multivariable polynomials.
Understanding polynomials is essential since they are the building blocks for many algebraic concepts, including factoring quadratics and finding the domain of functions.
Factoring quadratics
Factoring quadratics is all about breaking down a quadratic expression into simpler pieces called factors. Quadratics are polynomials where the degree is 2, meaning the highest power of the variable is 2. Factoring allows us to see the roots or solutions of the equation more clearly.
Here's the process to factor a standard quadratic like \(ax^2 + bx + c\):
  • Identify a, b, and c in the quadratic expression.
  • Use factoring methods such as factoring by grouping, the quadratic formula, or special patterns like completing the square.
  • For example, if we have a quadratic \(c^2 - 5c - 36\), we can factor it into \((c - 9)(c + 4)\) as the original exercise shows.
The purpose of factoring is to simplify the problem, allowing you to find the zeroes of the polynomial or the critical points in a function.
Functions in algebra
Functions in algebra describe relationships between quantities where each input value (often represented as \(x\) or another variable) is related to exactly one output value (usually \(y\) or\(f(x)\)). This relationship can be expressed in different forms, like equations, graphs, or tables.
Key aspects of functions include:
  • The domain of a function is the complete set of input values where the function works. For a function to be defined, its calculations should not involve division by zero or other undefined operations.
  • To determine the domain, consider any restrictions on the input caused by square roots, denominators, and logarithms. For example, in this case, the function \(r(c) = \frac{c+3}{c^2 - 5c - 36}\) is undefined where \(c^2 - 5c - 36 = 0\).
  • The range is the set of possible output values, but in this problem, focusing on the domain is key.
Understanding functions is vital as they model real-world phenomena, like the cost of goods, and solve various practical and theoretical problems in math.

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Most popular questions from this chapter

For each pair of functions, find a) \(\left(\frac{f}{g}\right)(x)\) and b \(\left(\frac{f}{g}\right)(-2)\) Identify any values that are not in the domain of \(\left(\frac{f}{g}\right)(x)\). $$f(x)=x^{2}-15 x+54, g(x)=x-9$$

Let \(f(x)=-x^{2}+10 x+4\) and \(g(x)=x+1 .\) Find a) \((g \circ f)(x)\) b) \(\quad(f \circ g)(x)\) c) \((f \circ g)(-2)\)

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=x^{2}\) is reflected about the \(x\) -axis.

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=x^{2}\) is shifted right 5 units and down 1.5 units.

Graph the following greatest integer functions. $$h(x)=[x-1]$$
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