91Ó°ÊÓ

Function notation is a way to name a function that is simple and helps you understand what is being done in the problem. Imagine a function as a machine. You put something in, the machine does some work, and you get something out.

For example, with the function notation \(f(x)\), the \(x\) is what you put into the machine. The function, \(f\), describes what the machine does. In our case, it tells us it takes \(x\) and transforms it.

So, if \(f(x) = k x^{-1}\), it means you input \(x\) and the function gives you \(k \cdot x^{-1}\). This tells us that to get the output, you must take \(x\) to the power of -1 and then multiply by \(k\).

When we write \(f(2x)\), we replace every \(x\) in the function definition with \(2x\). It’s like changing the ingredients of what you put into the machine. This helps us find out how the output changes with different inputs.

To summarize:

• \(f(x)\): input is \(x\)
• \(f(2x)\): input is \(2x\)
• \(f(3x)\): input is \(3x\)
• \(f(x/2)\): input is \(x/2\)

Inverse powers can be tricky but are quite simple once you understand the basics. An inverse power, like \(x^{-1}\), means you take the reciprocal of \(x\). For instance:

• \(x^{-1} = \frac{1}{x}\)
• \(x^{-2} = \frac{1}{x^2}\)

With \(f(x) = k x^{-1}\), when we change \(x\) to \(2x\), \(3x\), or \(\frac{x}{2}\), we are altering the value inside the inverse power:

• \(f(2x) = k (2x)^{-1} = \frac{k}{2x}\)
• \(f(3x) = k (3x)^{-1} = \frac{k}{3x}\)
• \(f(\frac{x}{2}) = k (\frac{x}{2})^{-1} = \frac{2k}{x}\)

Similarly, for \(g(x) = k x^{-2}\), the squared inverse means it's even more reduced:

• \(g(2x) = k (2x)^{-2} = \frac{k}{4x^2}\)
• \(g(3x) = k (3x)^{-2} = \frac{k}{9x^2}\)
• \(g(\frac{x}{2}) = k (\frac{x}{2})^{-2} = \frac{4k}{x^2}\)

Inverse powers reduce the values significantly, especially for squared inverses.

Substitution is a fundamental concept in algebra where you replace a variable with another expression. This helps in finding specific values and simplifying functions.

In our problem, for each function (\(f\) and \(g\)), we are substituting different expressions for \(x\) to see how it changes the value of the function.

For example, starting with substitute inside the function:

• To find \(f(2x)\), replace \(x\) with \(2x\) in \(f(x)\), so it becomes \(f(2x) = k (2x)^{-1}\).
• To find \(f(3x)\), replace \(x\) with \(3x\) in \(f(x)\), so it becomes \(f(3x) = k (3x)^{-1}\).
• To find \(f(\frac{x}{2})\), replace \(x\) with \(\frac{x}{2}\), so it becomes \(f(\frac{x}{2}) = k (\frac{x}{2})^{-1}\).
• Similarly, for function \(g(x)\):
  • \(g(2x)\), replace \(x\) with \(2x\).
  • \(g(3x)\), replace \(x\) with \(3x\).
  • \(g(\frac{x}{2})\), replace \(x\) with \(\frac{x}{2}\).

Substitution is about changing the