91Ó°ÊÓ

A function is like a special machine in math that takes in a number and gives out another number based on some rules. In the exercise given, we have the function \( f(x) = 2 + 2 \sqrt{5-x} \). The part "\( f(x) \)" is like a placeholder for whatever input "x" you choose. The rule here is defined inside the equation. The function takes an "x", subtracts it from 5, finds the square root, multiplies it by 2, and finally adds 2 to the result.

Understanding how this function works is key to evaluating it at various points. When we're given specific values to plug into "x", like \(-4\), \(1\), or any expression like \(x+5\), we simply replace "x" in the function with the value given and follow the defined rules to compute it.

The substitution method is a basic but powerful tool in evaluating functions. The idea is straightforward: wherever you see the variable "x" in the function's formula, replace it with the given number or expression.

For example, in the solution for \( f(-4) \), every "x" in \( 2 + 2 \sqrt{5-x} \) was replaced with \(-4\), resulting in \( f(-4) = 2 + 2 \sqrt{5-(-4)} \). From there, you simplify step by step.

Using substitution ensures you follow the correct order of operations (PEMDAS/BODMAS) to evaluate exactly what the function outputs at specific values. It helps confirm understanding that the function's variable, "x", is just a placeholder for whatever numerical value you substitute into it.

Simplifying square roots is crucial when dealing with functions like \( f(x) = 2 + 2 \sqrt{5-x} \). Square root simplification often requires breaking down the square root to its simplest form.

Consider the expression \( \sqrt{9} \) which appears in the calculation, \( f(-4) \). Here, it's easy because 9 is a perfect square, and simplifying it gives 3 because \( 3 \times 3 = 9 \).

With examples like \( \sqrt{\frac{9}{4}} \), which appear in \( f(\frac{11}{4}) \), you simplify by finding the square root of the top and bottom separately: \( \frac{3}{2} \). Breaking down more complex roots into their simplest terms when you can makes the remainder of the calculation much easier.

Mathematical simplification involves breaking down expressions into their simplest form for easier calculation. It's about managing parts of an equation methodically to make them easier to handle and understand.

In the solutions offered, after substituting the values into \( f(x) \), simplification was used to break it down from an initial complex form to a simple number answer.

For example, in finding \( f(1) \), after substitution, the expression is \( 2 + 2 \times 2 \). Prioritize operations and simplify the arithmetic to get the final answer, \( 6 \). Simplifying handles each chunk of math so it fits together neatly and produces the right number without extra fuss.