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Chapter 2: Problem 5

In Exercises 1鈥�30, find the domain of each function. $$ f(x)=x^{2}-2 x-15 $$

Short Answer

Expert verified
The domain of the function is the set of all real numbers.

Step by step solution

01

Identify the type of function

The given function \(f(x) = x^{2}-2x-15\) is a quadratic function, specifically a polynomial function, which is defined for all real numbers.
02

Determine the domain

Since the function is a polynomial, the function is defined for all real numbers. Therefore, the domain of the function is all real numbers.

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Key Concepts

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

Quadratic Functions
Quadratic functions are a type of polynomial function. They are specifically characterized by the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
These functions form a parabolic shape on the Cartesian plane, which can open upwards or downwards depending on the sign of \(a\).
Here鈥檚 what you should know about quadratic functions:
  • A quadratic function is always continuous and smooth without any breaks or sharp corners.
  • Its graph is called a parabola.
  • The vertex of the parabola is the highest or lowest point, depending on the direction it opens.
  • The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
Understanding quadratic functions is essential as they appear frequently in various areas such as physics, engineering, and economics.
Polynomial Functions
Polynomial functions are sums of terms consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. They have the general form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\).
The highest power of the variable \(x\) determines the degree of the polynomial.
Some important characteristics of polynomial functions include:
  • They are defined for all real numbers, meaning their domain is always all real numbers.
  • Their graphs are smooth and continuous without any breaks.
  • The degree of the polynomial dictates the number of real roots, the basic shape of the graph, and the number of turning points.
  • Polynomial functions include linear functions (degree 1) and quadratic functions (degree 2), among others.
Knowing the nature of polynomial functions helps in understanding and solving various types of algebraic and calculus problems.
Real Numbers
Real numbers include all the numbers that can be found on the number line. This set encompasses both rational numbers (like 1/2, 4, or -7) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)).
Real numbers are used in measuring continuous quantities and are an essential part of mathematical analysis.
Here鈥檚 a snapshot of the properties of real numbers:
  • They are ordered: Any two real numbers can be compared to determine which is larger.
  • They include the integers, which are positive and negative whole numbers, and their zero.
  • Real numbers follow properties of addition and multiplication such as distributive, associative, and commutative properties.
  • Real numbers form a complete ordered field, meaning they include limits for sequences and series that converge.
In mathematics and applications, understanding real numbers is crucial as they serve as a foundation for more complex number systems and calculus concepts.

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Most popular questions from this chapter

Graph \(y_{1}=x^{2}-2 x, y_{2}=x,\) and \(y_{3}=y_{1} \div y_{2}\) in the same \([-10,10,1]\) by \([-10,10,1]\) viewing rectangle. Then use the TRACE \(]\) feature to trace along \(y_{3} .\) What happens at \(x=0 ?\) Explain why this occurs.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I have two functions. Function \(f\) models total world population \(x\) years after 2000 and function \(g\) models population of the world's more-developed regions \(x\) years after \(2000 .\) I can use \(f-g\) to determine the population of the world's less-developed regions for the years in both function's domains.

Express the given function \(h\) as \(a\) composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=|3 x-4|$$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{array}{r} {x^{2}+y^{2}=9} \\ {x-y=3} \end{array}$$

Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=\sqrt{x}, g(x)=x-1$$
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