Evaluate f(x)=2 square root of -9
The question asks to evaluate the function f(x) = 2√(-9) where "√" denotes the square root operation. Essentially, you are being asked to calculate the value of the function when x is substituted into the equation. However, the presence of the negative number (-9) under the square root raises the issue of finding the square root of a negative number, which is not possible within the set of real numbers and involves the concept of imaginary numbers in complex number theory.
$f \left(\right. x \left.\right) = 2 \sqrt{- 9}$
Answer
Solution:
Step 1: Simplify the expression $2\sqrt{-9}$.
Step 1.1: Express the function as $f(x) = 0 + 0 + 2\sqrt{-9}$.
Step 1.2: Combine like terms to simplify $f(x) = 2\sqrt{-9}$.
Step 1.3: Factor -9 as $-1(9)$ to get $f(x) = 2\sqrt{-1(9)}$.
Step 1.4: Separate the square root of the product into the product of square roots, yielding $f(x) = 2(\sqrt{-1} \cdot \sqrt{9})$.
Step 1.5: Replace $\sqrt{-1}$ with the imaginary unit $i$, so $f(x) = 2(i\sqrt{9})$.
Step 1.6: Express 9 as $3^2$ to get $f(x) = 2(i\sqrt{3^2})$.
Step 1.7: Extract the square root of the perfect square to obtain $f(x) = 2(i \cdot 3)$.
Step 1.8: Rearrange the terms to place the constant before the imaginary unit, resulting in $f(x) = 2(3i)$.
Step 1.9: Multiply 3 by 2 to finalize the simplification as $f(x) = 6i$.
Step 2: Isolate the function $f(x)$ on one side of the equation.
Step 2.1: Subtract $f(x)$ from both sides to get $0 = 6i - f(x)$.
Step 2.2: Rearrange the terms to form $0 = -f(x) + 6i$.
Step 3: Rewrite the equation in standard form.
Step 3.1: Present the equation as $-f(x) + 6i = 0$.
Knowledge Notes:
Imaginary Numbers: Imaginary numbers are an extension of the real number system where the square root of a negative number is defined. The imaginary unit is denoted as $i$, where $i^2 = -1$.
Square Roots: The square root of a number $x$ is a value that, when multiplied by itself, gives $x$. For positive real numbers, the square root is also real. However, for negative numbers, the square root is imaginary.
Simplification: Simplifying an expression involves combining like terms and reducing the expression to its simplest form. This can include factoring, expanding, and canceling terms.
Complex Numbers: Complex numbers consist of a real part and an imaginary part, expressed as $a + bi$ where $a$ and $b$ are real numbers. In this problem, the result is purely imaginary since the real part is zero.
Radical Properties: The square root of a product can be expressed as the product of the square roots of the factors. Additionally, the square root of a perfect square, such as $9$, is a rational number ($\sqrt{9} = 3$).
Equation Rearrangement: Rearranging an equation involves moving terms from one side of the equation to the other, often to isolate a variable or to express the equation in a standard form.
Multiplication of Imaginary Numbers: When multiplying an imaginary number by a real number, the result is an imaginary number. The multiplication follows the commutative property, allowing the rearrangement of terms.
