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Chapter 2: Problem 23

In Exercises 23-36, find the domain of the function. $$ f(x)=x^{2}+3 $$

Short Answer

Expert verified
The domain of the function \(f(x) = x^2 + 3\) is all real numbers, which can be written as \((-\infty, \infty)\) or \(\{x\;|\;x \in \mathbb{R}\}\).

Step by step solution

01

Identify any restrictions on the input values (x-values) of the function

Upon examining the function \(f(x) = x^2 + 3\), we can see that it is a quadratic function, and there are no restrictions on the input values; no denominators that can be zero, and no expressions inside square roots or logarithms that would generate undefined scenarios.
02

Write the domain

Since there are no restrictions on the input (x-values), the domain of the function is all real numbers. This can be written in interval notation as \((-\infty, \infty)\) or as a set notation: \(\{x\;|\;x \in \mathbb{R}\}\).
03

State the final answer

The domain of the function \(f(x) = x^2 + 3\) is all real numbers, which can be written as \((-\infty, \infty)\) or \(\{x\;|\;x \in \mathbb{R}\}\).

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Key Concepts

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

Domain of a Function
The domain of a function is the set of all possible input values (typically referred to as "x" values) that a function can accept. Understanding the domain of a function is crucial because it tells us for which values the function is defined and can be calculated.For many functions, determining the domain requires identifying restrictions or limitations:
  • Denominators: If a function includes division by an expression, any values that make the denominator zero must be excluded from the domain.
  • Square roots: For functions with square roots, the expression inside must be non-negative to ensure the output remains a real number.
  • Logarithms: In logarithmic functions, the base must be positive and cannot be equal to one.
In our exercise, the function in question is a simple polynomial: \(f(x) = x^2 + 3\). This type of function does not have any of the restrictions mentioned above. Hence, the function is defined for all real numbers.
Quadratic Function
A quadratic function is a type of polynomial function with a degree of two. It can be generally expressed in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).Key characteristics of quadratic functions include:
  • Parabolic Shape: The graph of a quadratic function is a parabola, which may open upwards or downwards depending on the sign of \(a\).
  • Vertex: The highest or lowest point of a parabola, known as its vertex, provides valuable insights into the function's behavior.
  • Axis of Symmetry: A line that runs vertically through the vertex, dividing the parabola into two symmetrical halves.
In our example, \(f(x) = x^2 + 3\), the function is quadratic because it follows the structure of \(ax^2 + bx + c\) with \(a = 1\), \(b = 0\), and \(c = 3\). This particular quadratic is centered on the y-axis and shifts the entire graph upwards by three units, maintaining the parabolic shape.
Real Numbers
Real numbers constitute one of the most important sets in mathematics. They include:
  • Integers: Whole numbers and their negatives, like -3, 0, 4.
  • Fractions: Numbers that can be described as a ratio of integers, such as 1/2, 3/4.
  • Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or \(\sqrt{2}\).
  • Whole and Natural Numbers: Subsets of integers consisting of non-negative numbers.
Real numbers can be visually represented on a continuous number line, which captures all the points extending indefinitely in both directions.In the context of functions, when we say the domain includes all real numbers, it indicates that you can input any real number into the function, and it will produce a valid output. This is the case for the quadratic function \(f(x) = x^2 + 3\), as it accepts any real number as its input, meaning its domain stretches across the entire set of real numbers, \((-\infty, \infty)\).

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