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Understanding trigonometric function substitution can simplify solving tasks involving trigonometric identities. It's a handy tool when you need to solve or simplify more complex expressions in trigonometry. While the given exercise doesn't directly involve trig functions, the substitution principle is similar to substituting variables in algebraic expressions.

Example: If you have the trigonometric function \({\sin}^2{x}+{\cos}^2{x}=1\), and you're asked to replace \(x\) with \(-x\), you'll use the substitution method.

• You start by understanding the initial function.
• Next, replace every instance of \(x\) with \(-x\).
• Lastly, use trigonometric identities to simplify, if necessary.

In many problems, just like in our given exercise, substituting negative values can help you identify patterns or simplify the solving process dramatically.

Simplifying algebraic expressions is a crucial skill in mathematics. It involves making an expression as simple as possible by combining like terms and, when possible, reducing fractions or factoring. Let's revisit our original exercise to understand it better.

We started with the function \(f(x) = 3x^2 - x\). To find \(f(-x)\), we substituted \(-x\) for every \(x\) in the function:

\({f(-x) = 3(-x)^2 - (-x) = 3x^2 + x}\).

See how the substitution simplified the expression?

• We've changed \(-x\) to \[x\] in \({(-x)^2}\), which resulted in a positive \({x^2}\).
• Then, the \({-(-x)}\) became \({x}\), simplifying the expression even further.

Simplifying algebraic expressions helps you verify your work and ensures that your answers are in their most reduced form.

The absolute value function is essential for handling distances and magnitudes, without regard to direction. It’s often used in various areas of mathematics to ensure positive outputs.

In our given exercise, we have \({k(x) = |x + 3|}\). Although we did not use this function, understanding its properties is helpful.

The absolute value \(|x + 3|\) ensures that the output is always non-negative, regardless of \(x\)'s sign.

• For \(x = 2, |x + 3| = |2 + 3| = 5\).
• For \(x = -5, |x + 3| = |-5 + 3| = |-2| = 2\).

When substituting values into functions with absolute values, it’s crucial to carefully handle the sign changes. This ensures your simplified expressions remain accurate, preserving their intended meanings.

Overall, mastering absolute values and their properties will make you confident in working with a wide range of mathematical problems.