Problem 46 - Algebra A Combined Function

Question

Without graphing, find the domain of each function. $$ h(x)=\sqrt{x-17}-3 $$

Step-by-Step Solution

Answer

The domain of \( h(x) = \sqrt{x - 17} - 3 \) is \([17, \infty)\).

Step 1: Identify the Function Type

The function given is \( h(x) = \sqrt{x - 17} - 3 \). It contains a square root operation, which means we need to ensure the expression inside the square root, \( x - 17 \), is non-negative for \( h(x) \) to be defined.

Step 2: Set the Inside of the Square Root Non-negative

Since the square root must be of a non-negative number, we set the equation inside the square root greater than or equal to zero: \( x - 17 \geq 0 \).

Step 3: Solve the Inequality

Solve the inequality \( x - 17 \geq 0 \) to find the values of \( x \) that satisfy this condition. Add 17 to both sides to obtain \( x \geq 17 \).

Step 4: Determine the Domain

The solution \( x \geq 17 \) indicates that the function \( h(x) \) is defined for all \( x \) that are greater than or equal to 17. Thus, the domain is \([17, \infty)\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values or expressions. They tell us which quantity is greater, smaller, or sometimes, if they can be equal. Inequalities use symbols like:

• < "less than"
• > "greater than"
• \(\leq\) "less than or equal to"
• \(\geq\) "greater than or equal to"

These symbols are crucial when determining the domain of functions, as they help in setting conditions for defining expressions. For the function \(h(x) = \sqrt{x - 17} - 3\), using inequalities ensures that the term inside the square root is always non-negative. This condition is given by the inequality \(x - 17 \geq 0\). Solving this inequality helps us find the range of values for which the function is defined.

Exploring the Square Root Function

A square root function is a type of function that involves the square root of a variable. It is generally expressed as \(f(x) = \sqrt{x}\). The crucial aspect of a square root function is that it only accepts non-negative inputs. This is because the square root of a negative number is not a real number.

With our function \(h(x) = \sqrt{x - 17} - 3\), it implies we are dealing with the square root of \(x - 17\).

• The function ensures the expression under the square root (\(x - 17\)) is always non-negative to maintain a real output.
• In simpler terms, \(x\) must be at least 17 for the square root to be valid.

Understanding the behavior and constraints of square root functions helps us find the domain easily, ensuring that all operations under the root are valid.

Solving Inequalities with Context

Solving inequalities involves finding all possible solutions for the variable that make the inequality true. Typically, it requires manipulation of the inequality in a similar way you would an equation, ensuring to comply with the properties of inequalities.

For \(x - 17 \geq 0\), we solve it like an equation:

• Add 17 to each side: \(x \geq 17\)

This result informs us that any \(x\) greater than or equal to 17 makes the inequality true. This solution represents the domain of the function \(h(x)\), indicating all values for which \(h(x)\) is defined. Practice in solving these problems can improve proficiency in finding domains and working within the constraints set by functions like the square root.