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Chapter 9: Problem 40

Find the domain of the given function algebraically. $$ f(x)=\sqrt{7 x+8} $$

Short Answer

Expert verified
The domain of the function is \([-\frac{8}{7}, \infty)\).

Step by step solution

01

Understand the Domain Objective

The domain of a function consists of all the possible x-values that will not lead to mathematically undefined situations in the function. For the function \( f(x) = \sqrt{7x + 8} \), we need the expression inside the square root (\(7x + 8\)) to be non-negative because the square root of a negative number is not defined in the set of real numbers.
02

Set the Inside of the Square Root to be Non-negative

To ensure that \(7x + 8\) is non-negative, we set up the inequality: \(7x + 8 \geq 0\). This inequality should be solved to find the values of \(x\) that satisfy it.
03

Solve the Inequality

Solve the inequality \(7x + 8 \geq 0\):1. Subtract 8 from both sides: \(7x \geq -8\).2. Divide both sides by 7: \(x \geq -\frac{8}{7}\).This gives us the set of x-values for which the expression inside the square root is non-negative.
04

Write the Domain in Interval Notation

Convert the inequality \(x \geq -\frac{8}{7}\) into interval notation. This means all x-values starting from \(-\frac{8}{7}\) to infinity. In interval notation, the domain is \([-\frac{8}{7}, \infty)\).

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Key Concepts

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

Square Root Functions
Square root functions have a specific rule: they can only accept non-negative values inside the root. This is because taking the square root of a negative number gives an undefined result in the set of real numbers. For the function \( f(x) = \sqrt{7x + 8} \), this means that the expression \( 7x + 8 \) must be greater than or equal to zero.
To check if a square root function is defined, follow these simple steps:
  • Identify the expression inside the square root.
  • Ensure that this expression is non-negative.
This helps you find the domain of the function, which includes all possible values of \( x \) that won't cause the function to be undefined.
Inequality Solving
Solving inequalities is a key skill when working with square root functions. When you have an inequality like \( 7x + 8 \geq 0 \), you need to find all \( x \)-values that satisfy this inequality. Here's how you can solve it step by step:
  • First, isolate the term with \( x \) by subtracting 8 from both sides: \( 7x \geq -8 \).
  • Next, divide each side by the coefficient of \( x \), which is 7, resulting in \( x \geq -\frac{8}{7} \).
Remember, the solution to this inequality gives all possible \( x \)-values that make the expression inside the square root non-negative.
Interval Notation
Interval notation is a compact way to express the set of solutions to an inequality. It shows all possible values within a specified range. For the inequality \( x \geq -\frac{8}{7} \), the solution is expressed as \([\,-\frac{8}{7}, \infty)\). Here鈥檚 how to interpret interval notation:
  • The square bracket \([\,-\frac{8}{7}\,\) indicates that \(-\frac{8}{7}\) is included in the set (a closed interval).
  • The parenthesis \(\infty)\) means infinity is not included as it's not a fixed value (an open interval).
Interval notation is useful because it's concise and clearly communicates the range where the function is defined.

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