Problem 1 - Thomas Calculus

Question

In Exercises \(1-6,\) find the domain and range of each function. $$ f(x)=1+x^{2} $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \([1, \infty)\)

Step 1: Identify the domain

The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function \(f(x) = 1 + x^2\), it includes all real numbers because you can square any real number and add 1 to it. Thus, the domain is all real numbers, expressed as \( (-\infty, \infty) \).

Step 2: Determine the behavior of the function

Understand that \(f(x) = 1 + x^2\) represents a parabola that opens upwards, since \(x^2\) is the dominant term and its coefficient is positive. The lowest point of the function, or its vertex, occurs at \(x=0\).

Step 3: Calculate the range

The range of a function is the set of all possible output values (y-values) the function can produce. Since the minimum value of \(x^2\) is 0 and occurs at \(x = 0\), the minimum value of \(f(x)\) is \(1 + 0 = 1\). As \(x\) increases or decreases, \(f(x)\) will become larger without bound. Therefore, the range of \(f(x)\) is \([1, \infty)\).

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards, depending on its equation. In our function, \(f(x) = 1 + x^2\), the parabola opens upwards. This is because the term \(x^2\) is positive.

• The vertex of the parabola is the point where it turns. For \(f(x) = 1 + x^2\), the vertex is at \((0,1)\) since this is the smallest value the function can take.
• The axis of symmetry is a vertical line that passes through the vertex. In this case, it is the line \(x = 0\).

Parabolas are symmetric, meaning both sides of the axis are mirror images. As \(x\) moves away from zero in either direction, \(f(x)\) increases, causing the parabola to rise. Recognizing these properties helps in identifying key features of quadratic functions and their transformations.

Real Numbers

Real numbers include every number you can think of. From positive and negative integers to fractions, and irrational numbers like \(\sqrt{2}\), all fall under the realm of real numbers.

The domain of the function \(f(x) = 1 + x^2\) is all real numbers \((-\infty, \infty)\). This is because you can substitute any real number for \(x\), compute \(x^2\), and add 1 to it.

• No restrictions exist, such as dividing by zero or taking the square root of a negative number, ensuring the function is defined for all real numbers.

Real numbers have an important role in defining both the domain and range for functions, so understanding them is key to mapping out potential outputs and inputs.

Quadratic Functions

Quadratic functions have the general formula \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). They form parabolas when graphed. In our case, \(f(x) = 1 + x^2\) is a simplified quadratic function where \(a = 1\), \(b = 0\), and \(c = 1\).

Quadratic functions have distinct characteristics:

• A vertex, which is the highest or lowest point depending on whether the parabola opens up or down. \(f(x) = 1 + x^2\) has a vertex at \((0,1)\).
• Simplicity in calculation: Squaring any real number gives a non-negative result, influencing the range.

The range of the function is \([1, \infty)\) because the lowest value of \(f(x)\) is \(1\), occurring at \(x = 0\). The value of \(f(x)\) only increases as \(x\) changes.
Understanding quadratic functions and their graph behavior helps in analyzing and predicting the function's domain and range.