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Chapter 2: Problem 57

Find the domain of each function. $$f(x)=\frac{2}{(x+3)(x-7)}$$

Short Answer

Expert verified
The domain of the function \(f(x) = \frac{2}{(x+3)(x-7)}\) is all real numbers except -3 and 7. It can be described in interval notation as \( (-\infty, -3) \cup (-3, 7) \cup (7, +\infty) \).

Step by step solution

01

Identify the Denominator

The function in question is a rational function, and it's written in the form \(f(x) = \frac{A}{B}\), where both A and B are polynomials. Looking at our function, we seek to find the domain, and we notice that the denominator (B) is equal to \((x+3)(x-7)\). Remember, the function is undefined where the denominator equals zero.
02

Solve for x-values that make the denominator zero

To find the values that will make the denominator zero, we solve for x in the denominator equation. That is, we solve the equations \(x+3=0\) and \(x-7=0\). These are obtained from each factor in our denominator. Solving for x, we get x = -3 and x = 7.
03

Define the domain of the function

The domain of this function will be all real numbers except -3 and 7 because these values make the denominator equal to zero and thus make the function undefined. Therefore, our domain can be described in interval notation as \( (-\infty, -3) \cup (-3, 7) \cup (7, +\infty) \).

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Key Concepts

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

Rational Functions
Rational functions are a type of function expressed as the ratio of two polynomials. It can be represented in the general form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
The most important thing to note about rational functions is that they are only defined when the denominator \( Q(x) \) is not equal to zero. This is because division by zero is undefined in mathematics.
In the case of our function \( f(x) = \frac{2}{(x+3)(x-7)} \), the numerator is 2, and the denominator is \( (x+3)(x-7) \). The domain of the function will include all real numbers except those which make the denominator equal to zero.
When dealing with rational functions:
  • Check the denominator to identify values that make it zero.
  • Exclude these values from the domain.
Undefined Expressions
An expression is considered undefined for certain values when it involves division by zero. In mathematics, division by zero doesn't have a meaning and is therefore considered undefined.
In the context of the given rational function \( f(x) = \frac{2}{(x+3)(x-7)} \), we must determine which values of \( x \) will make \((x+3)(x-7) = 0\). This implies finding where each factor inside the denominator equals zero.
By setting \( x+3 = 0 \) and \( x-7 = 0 \), we can solve to find \( x = -3 \) and \( x = 7 \). These specific x-values make the expression undefined.
  • Identify expressions in the denominator that cause division by zero.
  • Solve each expression for zero to find the problem points.
Interval Notation
Interval notation is a concise way to describe subsets of the real number line. It is especially useful for defining the domain of functions such as rational functions, where certain x-values are excluded.
In interval notation:
  • Parentheses \( () \) indicate that the endpoint is not included in the interval.
  • Brackets \( [] \) would mean that an endpoint is included, but they are not used here because our endpoints correspond to undefined values (\( x = -3 \) and \( x = 7 \)).
Using interval notation, the domain of the function \( f(x) = \frac{2}{(x+3)(x-7)} \) is written as \((-\infty, -3) \cup (-3, 7) \cup (7, \infty)\). This notation shows the parts of the number line where the function is defined, concatenated with the union \( \cup \) symbol, which represents a collection of separate intervals.

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Most popular questions from this chapter

The number of lawyers in the United States can be modeled by the function $$ f(x)=\left\\{\begin{array}{ll} 6.5 x+200 & \text { if } 0 \leq x<23 \\ 26.2 x-252 & \text { if } x \geq 23 \end{array}\right. $$ where \(x\) represents the number of years after 1951 and \(f(x)\) represents the number of lawyers, in thousands. In Exercises \(85-88,\) use this function to find and interpret each of the following. $$ f(50) $$

What must be done to a function's equation so that its graph is shifted vertically upward?

Almanacs, newspapers, magazines, and the Internet contain bar graphs and line graphs that describe how things are changing over time. For example, the graphs in Exercises \(79-82\) show how various phenomena are changing over time. Find a bar or line graph showing yearly changes that you find intriguing. Describe to the group what interests you about this data. The group should select their two favorite graphs. For each graph selected: a. Rewrite the data so that they are presented as a relation in the form of a set of ordered pairs. b. Determine whether the relation in part (a) is a function. Explain why the relation is a function, or why it is not.

Which one of the following is true? a. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of three units to the right and three units upward of the graph of \(f\) b. If \(f(x)=-\sqrt{x}\) and \(g(x)=\sqrt{-x},\) then \(f\) and \(g\) have identical graphs. c. If \(f(x)=x^{2}\) and \(g(x)=5\left(x^{2}-2\right),\) then the graph of \(g\) can be obtained from the graph of \(f\) by stretching \(f\) five units followed by a downward shift of two units. d. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting in the \(x\) -axis, and then moving the resulting graph down four units.

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. On the other hand, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.
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