Question

Find the domain of each function. \(G(x)=\sqrt{1-x}\)

Short answer

Domain: \((-\infty, 1]\).

Step-by-step solution

Understand the Function

Observe that the function is given as \[G(x)=\sqrt{1-x}\]. The square root function is only defined for non-negative values.

Set Up the Inequality

To find the domain, set up the inequality by ensuring the term inside the square root is non-negative. Write: \[1-x \geq 0\].

Solve the Inequality

Solve the inequality derived in Step 2:\[1 - x \geq 0\] Subtract 1 from both sides: \[-x \geq -1\] Multiply both sides by -1 (remember to flip the inequality sign): \[x \leq 1\].

Write the Domain

The domain of the function is all x-values that satisfy the inequality \[x \leq 1\]. That can be written in interval notation as \((-\infty, 1]\).

Key concepts

inequalities

Inequalities are a fundamental concept in mathematics, used to compare the relative size of two values. In this problem, we work with the inequality long as the value on the right side is greater than or equal to the value on the left side. For example, . Hence, the inequality must be reversed. This new inequality must be solved carefully to ensure it maintains its properties.
When solving an inequality, it is vital to perform the same mathematical operations on both sides. This ensures equality is maintained. Remember always to flip the inequality sign when multiplying or dividing both sides by a negative number. By doing so, you keep the inequality true to its definition.

square root function

The square root function is one of the most common functions in mathematics, symbolized by evaluated at a This property is why we set up the inequality in the problem: to find where the expression inside the square root is greater than or equal to zero. Only when .
By doing so, we make sure that we are working within the real number system and keeping our calculations valid. These calculations will lead us to the function's domain by identifying which values of The values are allowed.

interval notation

Interval notation is a concise way of writing the domain and range of a function. In this notation, an interval is described using parentheses ( ) and brackets [ ].
Parentheses, ( or ), denote that the endpoint values are not included in the interval. While brackets, [ or ], indicate they are included. For instance:

• (a, b) means all numbers between a and b, but not including a and b.
• [a, b] includes all numbers between a and b, including a and b.

In our problem, we use interval notation to state the domain of
Intervals are a neat way to present domains and ranges of functions because they allow for clear and direct communication of which values a function can take. This simplifies understanding and solving more complex problems. Practicing with interval notation enhances comprehension in areas such as calculus and higher algebra.
Being comfortable with interval notation aids significantly in dealing with more advanced concepts down the line.