Functions 2

<a href="https://www.prepswift.com/quizzes/quiz/prepswift-functions-2" target="_blank">Functions 2 Exercise</a>This is called Functions II because there are two more complex things we need to consider. The Input Does Not Always Equal $x$ The vast majority of function problems look like $f(x)$ or $g(x)$ or something like that. Notice how you only see $x$ in the parentheses. However, you might encounter something like this: $$f(x+5) = x^3 - x^2$$ In this case, the input is not simply $x$, but rather $x+5$. The problem then might ask you what $f(6)$ equals. In that case, you have to follow these steps: Step 1: Set $x+5$ equal to $6$ to get $x=1$. Step 2: Plug the $x=1$ into the function's output equation $x^3-x^2$ to get $1-1=0$. Functions Equal to Other Functions We briefly addressed this in the mountain entry for Functions I. Let's dive more into it here. These problems can get quite complex. For example... $$f(2x) = (f(x))^2$$ What does this mean? If you take a look at the left side, notice how we're doubling the input with $2x$. And notice that, when we do that, the output gets squared. Here are some more examples: $$f(\frac{x}{3}) = \sqrt{f(x)}$$ Translation: When we divide the input by $3$, we take the square root of the output. $$f(x^2) = (f(x))!$$ Translation: When we square the input, we take the factorial of the output. Example Problem  If $f(x+7) = 2f(x)$, and $f(3) = 10$, what is $f(17)$? To solve this, first translate the algebra: when we add $7$ to the input, we double the output. $$f(3+7) = 2 \times 10$$ $$f(10) = 20$$ We repeat the process: $$f(10+7) = 2 \times 20$$ $$f(17) = 40$$