Question

Find the domain of \(f(x)=-9 \sqrt{x-4}+1\)

Short answer

The domain is \( x \geq 4 \).

Step-by-step solution

Identify the function's type

The function given is a square root function: \( f(x) = -9 \sqrt{x - 4} + 1 \). The expression inside the square root (\( x-4 \)) must be non-negative because the square root is only defined for non-negative arguments.

Set up the inequality

To find the domain, set the expression inside the square root greater than or equal to zero: \( x-4 \geq 0 \).

Solve the inequality

Solve the inequality for \( x \): \( x \geq 4 \).

Express the domain

The solution \( x \geq 4 \) represents the domain of the function. The function is defined for all values of \( x \) starting from 4 and extending to positive infinity.

Key concepts

square root function

Square root functions are mathematical functions of the form \(\f(x) = \sqrt{x} \).
They can include additional terms within the square root or outside of it, such as multiplying by a constant or adding/subtracting another number.
In our case, the function is \(\f(x) = -9 \sqrt{x - 4} + 1\).
The square root function requires that the argument inside the square root (here, \(x - 4\)) must be greater than or equal to zero because the square root of a negative number is not a real number.
Understanding square root functions is crucial to determine their domains and behavior.

inequality

Inequality is a mathematical statement indicating that two expressions are not equal and involves the symbols \(>\),\( \geq\),\( <\), or \( \leq\).
In solving square root functions, you'll often set up an inequality to ensure the expression inside the square root is non-negative.
For our function, we create the inequality \( x - 4 \geq 0 \) to find the values of \( x \).
This step helps find where the function is defined.
Approaching inequality requires a good grasp of basic algebraic manipulations and principles.

function domain

The domain of a function is the set of all possible input values (usually denoted as \( x \)) that the function is defined for.
For square root functions, ensuring that the value inside the root is non-negative is key to determining the domain.
By solving the inequality \( x - 4 \geq 0 \), we recognize the function \( f(x) = -9 \sqrt{x - 4} + 1 \) is defined starting from \( x = 4 \) and extending to positive infinity.
Therefore, the domain is \( x \geq 4 \), meaning all numbers 4 and greater are valid inputs.
Understanding the domain helps in graphing the function and predicting its behavior in different intervals.

solving inequalities

Solving inequalities involves finding the values of \( x \) that make the inequality true.
Here, we had to solve \( x - 4 \geq 0 \).
To do this, you simply add 4 to both sides to isolate \( x \), resulting in \( x \geq 4 \).
Solving inequalities often involves similar steps of isolating \( x \) through basic algebraic operations.
It's important to remember that if you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip.
In our given problem, no such step was necessary.