Question

\(g(x)=\sqrt{x}-5\)

Short answer

To solve the function \(g(x) = \sqrt{x}-5\) for a particular input \(x\), one needs to calculate the square root of \(x\) and subtract \(5\) from the result.

Step-by-step solution

Understand the Function

We have been given a function \(g(x) = \sqrt{x}-5\). This function will take a real number \(x\) as input and will output the value of \(\sqrt{x}-5\). Please note that the function is only defined for \(x\) values that are zero or positive, as we cannot take square roots of negative numbers.

Apply Value

Apply the desired value of \(x\) into the equation. Replace \(x\) to find the result.

Solve the Function

Calculate the square root of \(x\) and subtract \(5\) from the result to get the output of function \(g\).

Key concepts

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because when 3 is multiplied by itself (i.e., 3 * 3), it equals 9. Square roots are part of basic mathematical operations, often associated with radical expressions. The square root symbol is written as \( \sqrt{} \), and this operation is crucial for solving many algebraic equations.
Understanding square roots is significant when dealing with exponential growth, geometry, and various real-world applications. It's important to remember that square roots are only defined for non-negative numbers in the real number system because no real number squared will give a negative number. This means expressions like \( \sqrt{-1} \) are not defined within the scope of real numbers. In equations, square roots are frequently used to "undo" squaring an operation and to find the root value in various algebraic contexts.

Domain of a Function

The domain of a function refers to the set of all possible input values (usually \(x\)) that will give a valid output in the function. Identifying the correct domain is essential, as it prevents errors when evaluating a function. When we talk about domains, we're essentially asking: "For which values of \(x\) is this function defined?"
For the function \(g(x) = \sqrt{x}-5\), the domain is restricted to all real numbers \(x\) where the square root is defined. Since square roots of negative numbers aren't defined in the real number system, the domain of \(g(x)\) is restricted to \(x \geq 0\).
This means that the function can take any non-negative value of \(x\), starting from zero and extending to all positive numbers. Understanding domains helps in graphing functions and solving equations as it tells us which parts of the graph are valid or where we can expect to find real solutions.

Function Evaluation

Function evaluation involves computing the result of a function for given input values. It's like a machine where you input a number, and the function processes this number through its operations to produce an output.
To evaluate a function such as \(g(x) = \sqrt{x} - 5\), you simply substitute the input value you wish to evaluate into the function, replacing \(x\). Here's how:

• Choose a value for \(x\). Ensure it's within the domain of the function.
• Substitute this value into the function's equation.
• Perform the operations as indicated to get the result.

For example, if you need to find \(g(4)\), you substitute \(4\) into the function: \(g(4) = \sqrt{4} - 5\). Then calculate \(\sqrt{4} = 2\) and finally do the subtraction: \(2 - 5 = -3\). This process ensures accurate computations when dealing with complex or multiple functions, providing a systematic approach to obtain the output.