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Chapter 1: Problem 71

In Exercises 71-82, find the domain of the function. \( f(x) = 5x^2 + 2x - 1 \)

Short Answer

Expert verified
The domain of the function \( f(x) = 5x^2 + 2x - 1 \) is all real numbers.

Step by step solution

01

Identify the Function Type

Identify the type of function. In this case, the function \( f(x) = 5x^2 + 2x - 1 \) is a quadratic function. Quadratic functions are in the form of \( ax^2+bx+c \). Observing the function, there are no restrictions such as division by zero or square root of a negative number.
02

Decide Domain of the Function

Since there are no restrictions in the function, and quadratic functions can accept any real numbers, conclude that the domain of this 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
A quadratic function is a type of polynomial function specifically of degree 2. It is typically written in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

The graph of a quadratic function is a parabola, which can open either upwards or downwards depending on the sign of \( a \). If \( a > 0 \), the parabola opens upward, resembling a U-shape, and if \( a < 0 \), it opens downward, like an upside-down U.

Some key features of quadratic functions include:
  • Vertex: The point where the parabola changes direction.
  • Axis of symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • Roots or zeros: The x-values where the parabola intersects the x-axis.
This function type is crucial in various real-world applications, such as calculating areas, determining velocities, and modeling projectile motion.
Restrictions on Domain
The domain of a function refers to all the possible input values \( x \) for which the function is defined. When determining the domain, it is important to identify any restrictions that would make the function undefined.

Common restrictions arise in functions that include:
  • Division by zero, as it is undefined.
  • Square roots of negative numbers, which result in non-real numbers when dealing with real functions.
Quadratic functions, like \( f(x) = 5x^2 + 2x - 1 \), generally do not have these restrictions because:
  • There are no denominators, so division by zero is not a concern.
  • There are no square root operations involved.
Therefore, quadratic functions typically have a domain of all real numbers, as they can take any real number as an input.
Real Numbers
Real numbers encompass all the numbers that can be found on the number line. This includes both rational numbers (such as fractions and whole numbers) and irrational numbers (like the square root of 2 or \( \pi \)). Real numbers are fundamental in mathematics because they are used in a wide array of calculations and problem-solving scenarios.

Key features of real numbers include:
  • They can be positive, negative, or zero.
  • They can be expressed as decimals, even if the decimal representation is infinitely long.
  • They form a complete ordered field, meaning any two real numbers can be compared, and there is a real number between any two distinct real numbers.
When discussing the domain of a quadratic function, the phrase "all real numbers" indicates that any real number is permissible as an input value for the function. This all-encompassing nature is why quadratic functions have such a broad application in various fields.

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Most popular questions from this chapter

In Exercises 67-74, find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(z\) varies jointly as \(x\) and \(y\). (\(z = 64\) when \(x = 4\) and \(y = 8\).)

NEWTON'S LAW OF UNIVERSAL GRAVITATION: The gravitational attraction \(F\) between two objects of masses \(m_1\) and \(m_2\) is proportional to the product of the masses and inversely proportional to the square of the distance \(r\) between the objects.

The direct variation model \(y = kr^n\) can be described as "\(y\) varies directly as the \(n\)th power of \(x\)," or "\(y\) is ________ ________ to the \(n\)th power of \(x\)."

RESISTANCE In Exercises 77 and 78, use the fact that the resistance of a wire carrying an electrical current is directly proportional to its length and inversely proportional to its cross-sectional area. If #28 copper wire (which has a diameter of 0.0126 inch) has a resistance of 66.17 ohms per thousand feet, what length of #28 copper wire will produce a resistance of 33.5 ohms?

In Exercises 27-30, use the given value of \(k\) to complete the table for the inverse variation model \(y = \frac{k}{x^2}\) Plot the points on a rectangular coordinate system. \(k = 5\)
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