Advanced Guessing Strategies

As you progress to harder difficulties, you won't just see simple functions like $\sin(x)$ or $e^x$. You will encounter combinations of functions. To solve these, you need to learn how to deconstruct them.

1. Function Arithmetic ($f(x) + g(x)$)

When two functions are added, their graphs "stack" on top of each other. The most common pattern is adding a "trend" to a "detail".

Example: The Wobbly Line

Consider the function: $f(x) = x + \sin(2x)$

This is a sum of:

1. Trend: $y = x$ (A diagonal line)
2. Detail: $y = \sin(2x)$ (A wave)

At a glance, it looks like a diagonal line that wiggles.

💡 The Decomposition Strategy

In Function Guessr, you can use the graphing tool to help you think. If you suspect a function follows a certain trend (like $y=x$), you can subtract that trend to see what remains.

Step-by-Step Deduction:

1. Observe: "The graph generally goes up like $y=x$, but it's wavy."
2. Hypothesis: "Maybe it's $x + \text{something}$."
3. Test: Type f(x) - x into the input box to see the "residual" graph.

If you subtract $x$ from $x + \sin(2x)$, you get just $\sin(2x)$:

Now the problem is reduced to identifying a simple sine wave!

Pro Tip: Always look for the dominant behavior first.

• If it shoots up like a parabola ($x^2$), try looking at $f(x) - x^2$.
• If it grows exponentially, try dividing: $f(x) / e^x$.

---

2. Composite Functions ($f(g(x))$)

Composite functions wrap one function inside another.

Example: The Bouncing Wave

$f(x) = |\sin(x)|$

Here, the inner function is $\sin(x)$ and the outer function is $|x|$ (Absolute Value).

• $\sin(x)$ goes up and down between -1 and 1.
• $|x|$ effectively "folds" the negative parts up.
• Result: A wave that bounces off the x-axis, never going below zero.

Example: The Frequency Shift

$f(x) = \sin(x^2)$

Here, the inner function is $x^2$. As $x$ gets larger, $x^2$ grows very fast.

• This causes the input to the sine function to change faster and faster.
• Result: A wave that oscillates more and more frantically as you move away from zero.

---

3. Summary of Operations