Question

a) Identify the domains and ranges of $$y=x^{2}-4 \text { and } y=\sqrt{x^{2}-4}$$. b) Why is \(y=\sqrt{x^{2}-4}\) undefined over an interval? How does this affect the domain of the function?

Short answer

Domains: \((-\infty, \infty)\) for \( y = x^2 - 4 \) and \((-\infty, -2] \cup [2, \infty)\) for \( y = \sqrt{x^2 - 4} \). Undefined because \( x^2 - 4 \) must be non-negative.

Step-by-step solution

Identify the Domain of y = x^2 - 4

For the function \( y = x^2 - 4 \), the domain is all real numbers since there are no restrictions on the values of \( x \). This is because the expression \( x^2 - 4 \) is defined for all \( x \in \mathbb{R} \). So, the domain is \( (-\infty, \infty) \).

Identify the Range of y = x^2 - 4

Next, determine the range of \( y = x^2 - 4 \). The function \( x^2 - 4 \) opens upwards (U-shape) and its vertex is at the minimum point \( (0, -4) \). Thus, the function can take any value greater than or equal to -4. Therefore, the range is \( [-4, \infty) \).

Identify the Domain of y = \sqrt{x^2 - 4}

For the function \( y = \sqrt{x^2 - 4} \), the expression inside the square root, \( x^2 - 4 \), must be non-negative (\( x^2 - 4 \geq 0 \)). This implies \( x^2 \geq 4 \) which simplifies to \( x \geq 2 \) or \( x \leq -2 \). Thus, the domain is \( (-\infty, -2] \cup [2, \infty) \).

Identify the Range of y = \sqrt{x^2 - 4}

Since \( y = \sqrt{x^2 - 4} \) represents the principal (non-negative) square root, the smallest value it can achieve is 0 (when \( x = \pm 2 \)). As \( x \) increases or decreases beyond \( \pm 2 \), the value of \( \sqrt{x^2 - 4} \) increases without bound. Therefore, the range is \( [0, \infty) \).

Explain Why y = \sqrt{x^2 - 4} is Undefined Over an Interval

The function \( y = \sqrt{x^2 - 4} \) is undefined for values of \( x \) such that \( -2 < x < 2 \) because within this interval, \( x^2 - 4 \) is negative, leading to an undefined square root. As a result, the domain does not include these values of \( x \).

Key concepts

Domain of Functions

In mathematics, the *domain of a function* refers to the set of all possible input values (usually denoted as \( x \)) that the function can accept without resulting in an undefined expression. For instance, in the function \( y = x^2 - 4 \), any real number \( x \) can be substituted into the function, causing its domain to be all real numbers, which is denoted as \( (-fty, fty) \). In contrast, the function \( y = \text{sqrt}(x^2 - 4) \) has restrictions. Inside the square root, the expression \( x^2 - 4 \) must be non-negative to avoid taking the square root of a negative number, which is not defined in the real number system. Therefore, this limits the domain to \( x \text{ such that } x geq 2 \text{ or } x leq -2 \), resulting in the domain \( (-fty, -2] \cup [2, fty)\).

Range of Functions

The *range of a function* is the set of all possible output values (often denoted as \( y \)) that result from using all the input values in the domain. Taking the function \( y = x^2 - 4 \) as an example, since \( x^2 \) is always non-negative and reaches its minimum value at \( x = 0 \), the range starts from \( -4 \) as the lowest value (when \( x = 0 \)), and extends infinitely upwards. As a result, the range is \( [-4, fty) \). For the square root function \( y = \text{sqrt}(x^2 - 4) \), the output is non-negative by definition. The smallest value occurs when \( x^2 - 4 = 0 \) (i.e., when \( x = 2 \text{ or } -2 \)), giving the minimum value 0, and as \( x \) increases or decreases beyond \( \text{ pm } 2 \), the value of \( y \) increases without bound. Hence, the range is \( [0, fty) \).

Square Root Functions

Square root functions involve finding a number that, when multiplied by itself, returns the given value. Symbolically, for the function \( y = \text{sqrt}(x^2 - 4) \), it seeks the non-negative root of the expression inside the square root. This function is defined only for values of \( x \) where \( x^2 - 4 \geq 0 \). Simplifying, this inequality results in \( x \geq 2 \text{ or } x \leq -2 \). Therefore, the domain of this function consists of intervals where the expression inside the square root is non-negative. For example, if \( x \) is within the interval \((-2, 2)\), the expression inside the square root is negative, leading to an undefined square root in the real number system.

Quadratic Functions

A quadratic function is a polynomial function of the form \( y = ax^2 + bx + c \), where \( a eq 0 \). The simplest quadratic functions, like \( y = x^2 - 4 \), create a parabolic graph that opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). For the function \( y = x^2 - 4 \), the vertex or the lowest point is at \( (0, -4) \). Due to the nature of the parabola which opens upwards, there is no restriction on \( x \) values, making the domain \( (-fty, fty) \). The range, however, is restricted, starting from the vertex at \( -4 \) to positive infinity, giving the range \( [-4, fty) \). Quadratic functions are foundational in algebra and appear in various contexts ranging from physics to economics.