Problem 12 - Algebra Form and Function

Question

Find (a) The domain. (b) The range. $$ y=\sqrt{2 x-4} $$

Step-by-Step Solution

Answer

Answer: The domain of the function is \(x \in [2, \infty)\) and the range is \(y \in [0, \infty)\).

Step 1: Find the domain

To find the domain of the function, we need to determine the values of x for which the square root function is defined. Since the square root of a negative number is not real, we need to find the values of x for which the expression inside the square root (the radicand) is non-negative. So, let's find the values for which: $$ 2x-4 \ge 0 $$ Add 4 to both sides: $$ 2x \ge 4 $$ Divide by 2: $$ x \ge 2 $$ The domain of the function is: $$ x \in [2, \infty). $$

Step 2: Find the range

For the range of the function, we need to determine the possible values of y, given the domain found in Step 1. Given that x takes on values in the interval [2, ∞), we can check how y behaves along this interval: For x = 2: $$ y = \sqrt{2(2) - 4} = \sqrt{0} = 0 $$ As x increases, so does the value of the radicand \(2x - 4\), resulting in an increasing value of y. Since there is no upper bound for x, there is also no upper bound for y. Thus, the range of the function is: $$ y \in [0, \infty). $$ So, the domain and range of the function y = √(2x - 4) are: Domain: $$x \in [2, \infty)$$ Range: $$y \in [0, \infty)$$

Key Concepts

square root function

The square root function, represented as \( y = \sqrt{x} \), is an essential concept in algebra that transforms a non-negative input into its square root. But what exactly does this mean in terms of real numbers? Square roots essentially "reverse" the process of squaring a number. For a number \( a \), the square root \( \sqrt{a} \) is the non-negative value that, when multiplied by itself, equals \( a \). This is why the square root of a number, such as 9, is 3 (since \( 3 \times 3 = 9 \)).

In the context of functions like \( y = \sqrt{2x - 4} \), the function only takes non-negative inputs for the expression inside the square root (known as the radicand). This is because the square root of a negative number is not a real number. When working with square root functions, we often analyze two facets: the domain and the range. The domain refers to input values (\( x \)) that do not make the radicand negative, and the range signifies possible output values (\( y \)).

With \( y = \sqrt{2x - 4} \), understanding square roots helps us establish the boundaries within which the function operates, ensuring that we only handle real numbers.

inequality solving

Solving inequalities is a crucial skill in algebra, especially when determining the domain of functions like square root functions. Inequalities help determine the set of values that form the function’s domain by finding solutions where an equation or expression remains valid. For instance, if we consider \( 2x - 4 \geq 0 \), it sets up a condition to avoid negative values under the square root

Let's break down the inequality \( 2x - 4 \geq 0 \):

• First, isolate terms involving \( x \) by adding 4 to each side, resulting in \( 2x \geq 4 \).
• Next, solve for \( x \) by dividing both sides by 2 to maintain balance in the inequality, giving us \( x \geq 2 \).

This solution indicates that \( x \) can be any value greater than or equal to 2. Thus, solving inequalities is foundational in defining limitations of expressions and ensuring meaningful, real answers.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They are the backbone of algebra and are crucial in solving real-world problems. Expressions like \( 2x - 4 \) in the function \( y = \sqrt{2x - 4} \) demonstrate their utility in transforming and analyzing relationships mathematically.

In an algebraic expression, every part has a role:

• Constants, like 4 in \( 2x - 4 \), provide fixed values that shift or translate the graph.
• Coefficients, such as 2 before \( x \), magnify or reduce the effect of the variable on the expression.
• Variables, represented by \( x \), allow flexibility, handling unknowns or ranges of values.

Understanding algebraic expressions involves recognizing how each component influences the whole. They let us model situations abstractly, predictably analyzing anything from simple profit calculations to complex scientific models.