Problem 67 - College Algebra

Question

Find the domain of each function. $$ g(x)=\sqrt{5 x+35} $$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \sqrt{5x+35}\) is \(x \geq -7\).

Step 1: Setting the expression under the square root greater or equal to zero

In order to find the domain, we need to set the expression under the square root, \(5x+35\), as greater or equal to zero, i.e., \(5x+35 \geq 0\).

Step 2: Solving the inequality

We will now solve the inequality for \(x\). To do this, subtract 35 from both sides: \(5x \geq -35\). Then, divide by 5 to solve for \(x\): \(x \geq -7\).

Step 3: Writing the Domain

After solving for \(x\), we can write the domain of the function. The domain of the function \(g(x)\) is all real numbers \(x\) such that \(x \geq -7\).

Key Concepts

Solving Inequalities

When it comes to solving inequalities, the goal is to find all the values of the variable that make the inequality true. An inequality is a mathematical statement that one quantity is less than or greater than another. They often appear in various mathematical contexts, and understanding how to solve them is critical for many functions, including finding domains.

The process for solving an inequality is similar to solving an equation: you perform operations to isolate the variable, keeping in mind that if you multiply or divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have an inequality like -2x > 4, and you divide by -2 to solve for x, the inequality becomes x < -2. After solving, it's important to express the solution as a set of numbers, which may be presented in interval notation, on a number line, or using set-builder notation.

Square Root Functions

A square root function is a type of radical function where the variable is under a square root. One key characteristic of square root functions is that the quantity under the square root, called the radicand, must be greater than or equal to zero since the square root of a negative number is not defined in the set of real numbers. This requirement directly influences the domain of the function.

To graph a square root function, you typically plot points by substituting values of the variable and finding their square roots. The graph starts at the point which makes the radicand zero and extends infinitely to the right, forming half of a parabola on its side. This shape reflects the fact that for every positive value inside the square root, there is a single positive square root. There are also functions involving the negative square root which reflect across the x-axis.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols, but no equals sign. For instance, 5x + 35 is an algebraic expression within the square root function in the given exercise. Expressions can be simplified or manipulated to solve for variables or to find specific values.

In algebra, you'll often factor expressions, combine like terms, or use various properties, such as distributive or associative, to rearrange and solve expressions. Understanding how to work with algebraic expressions is foundational to progressing in mathematics as they are the building blocks for forming equations and functions. 5x + 35 illustrates an expression where you can perform operations to solve for x, as was done in the exercise to find the domain of a function.