Problem 24 - Algebra for College Students

Question

\(f(x)=\sqrt{-1-x}\)

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, -1]\).

Step 1: Understand the Domain

The function is defined as the square root of \(-1 - x\). Since square roots are only defined for non-negative numbers, we set up the inequality \(-1 - x \geq 0\).

Step 2: Solve the Inequality

To find the values of \(x\) for which the expression under the square root is non-negative, solve \(-1 - x \geq 0\). This can be rewritten as \(-x \geq 1\).

Step 3: Isolate \(x\) in the Inequality

To isolate \(x\), divide both sides of the inequality by \(-1\), remembering to reverse the inequality sign. This gives \(x \leq -1\).

Step 4: Determine the Domain of the Function

The domain of the function \(f(x) = \sqrt{-1-x}\) is all real numbers \(x\) that satisfy \(x \leq -1\). Therefore, the domain of the function is \( (-\infty, -1] \).

Key Concepts

Understanding the Domain of a Function

When dealing with functions in algebra, determining the domain is crucial. The domain essentially tells us all the possible input values (x-values) that a function can accept and still produce a valid output. For the function given by \( f(x) = \sqrt{-1 - x} \), we must find the set of x-values where the expression under the square root remains non-negative.
The expression \(-1 - x\) must be greater than or equal to zero because square roots are only defined for non-negative numbers. Setting up the inequality \(-1 - x \geq 0\) helps us identify these values.

• Rewriting gives \(-x \geq 1\)
• Changing the inequality (divide by \(-1\)) flips the sign: \(x \leq -1\)

Thus, the domain of the function is all real numbers \(x\) satisfying \(x \leq -1\), which we express as \( (-\infty, -1] \).
Understanding this concept ensures that our function outputs real and valid numbers.

Solving Inequalities

Solving inequalities is a fundamental skill in algebra that helps us find the range of values a variable can take. In this particular problem, we encounter the inequality \(-1 - x \geq 0\). Here's how you can solve it step-by-step:

• First, isolate x by rewriting the inequality as \(-x \geq 1\).
• To make x the subject, divide both sides by \(-1\). Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.
• The inequality then becomes \(x \leq -1\).

This final inequality tells us that x should be less than or equal to \(-1\). Being comfortable with these rules helps solve various algebraic equations with ease, enhancing your problem-solving skills.

Exploring the Square Root Function

Square roots can initially seem tricky, but they become straightforward with a bit of practice. A square root function, like \( f(x) = \sqrt{-1 - x} \), involves finding a number which, when multiplied by itself, gives the expression under the square root.
The crucial part of any square root function is defining the domain, which involves ensuring the expression inside the square root is non-negative.

• In \( f(x) = \sqrt{-1 - x} \), we need \(-1 - x \geq 0\).
• This ensures the output stays within the real numbers.

If this condition isn't met, the resulting number becomes imaginary, which typically isn't usable in standard algebraic mathematics. Understanding how square root functions work allows you to handle complex functions and ensures your math skills remain sharp.