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Chapter 1: Problem 7

Find the domain of each function. $$g(x)=\frac{3}{x^{2}-2 x-15}$$

Short Answer

Expert verified
The domain of the function \(g(x)=\frac{3}{x^{2}-2x-15}\) is \(-\infty < x < -3 , -3 < x < 5 , 5 < x < \infty\).

Step by step solution

01

Identify the Denominator

In this function, the denominator is \(x^{2}-2x-15\)
02

Set the Denominator Equal to Zero

To find the x-values that cannot be in the domain, set the denominator equal to zero and solve for x: \(x^{2}-2x-15 = 0\)
03

Solve the Equation

This is a quadratic equation and can be factored to \((x-5)(x+3)=0\). Then, set each factor equal to zero and solve for x: \(x-5=0\) which gives \(x=5\) and \(x+3=0\) which gives \(x=-3\)
04

Identify the Domain

The domain of the function is all real numbers except for any values that make the denominator equal to zero. In this case, x cannot be 5 or -3. Therefore, the domain of g(x) is \(-\infty < x < -3\) or \(-3 < x < 5\) or \(5 < x < \infty\)

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Key Concepts

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

Factoring Quadratics
Quadratic equations take the form \( ax^2 + bx + c = 0 \). When faced with such an equation, factoring is one typical way to find solutions. Factoring involves expressing a quadratic as a product of binomials. For the quadratic \( x^2 - 2x - 15 \):
  • First, find two numbers that multiply to \( -15 \) (the constant term \(c\) ) and add up to \( -2 \) (the linear coefficient \(b\)).
  • Here, these numbers are \(-5\) and \(+3\) because \( (-5) \times (3) = -15 \) and \( (-5) + (3) = -2 \).
  • Use these numbers to rewrite the quadratic as a product of its factors: \((x - 5)(x + 3) = 0\).
Factoring is an efficient method because it breaks down equations into simpler components that are easier to solve. It's particularly useful for simplifying complex algebraic expressions.
Rational Functions
A rational function is any function that can be expressed as the quotient of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. For instance, the function given in the exercise, \( g(x)=\frac{3}{x^{2}-2x-15} \), is a rational function because it is in the form \( \frac{3}{\text{polynomial}} \). Understanding rational functions involves:
  • Recognizing that the value of these functions is undefined for values of \( x \) that make the denominator zero.
  • Analyzing how different values of \( x \) affect the overall value of the function.
Rational functions can have vertical asymptotes where the function heads towards infinity, or holes, which are points where the function is not defined.
Excluded Values
Excluded values arise when analyzing the domain of rational functions. These are the \(x\)-values that make the denominator zero and thus make the function undefined. To find them, take the following steps:
  • Identify the denominator of the rational function. In the exercise, \( x^2 - 2x - 15 \).
  • Set the denominator equal to zero, \( x^2 - 2x - 15 = 0 \), and solve for \( x \).
  • The solutions are \( x = 5 \) and \( x = -3 \).
These solutions correspond to the excluded values because in these cases, the function \( g(x) \) would require division by zero, which is undefined in mathematics. Thus, the domain of \( g(x) \) is all real numbers except for these excluded values.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a function to model data from 1990 through 2015 . The independent variable in my model represented the number of years after \(1990,\) so the function's domain was \(\\{x | x=0,1,2,3, \dots, 25\\}\)

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+x+y-\frac{1}{2}=0$$

Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\)

Perform the indicated operation or operations. $$(2 x-1)\left(x^{2}+x-2\right)$$
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