How do you describe the end behavior of common functions?

Expert intro: End behavior questions test whether you can read a graph or formula without overthinking it. The key is to track what happens to the output when the input becomes very large.

Detailed walkthrough

We describe each function in turn.

(i) $y=x^4$ as $x\to\infty$

As $x$ gets larger and larger, $x^4$ also gets larger and larger.

So

$y\to\infty.$

(ii) $y=(2-x)^3$ as $x\to\infty$

When $x$ becomes very large, $2-x$ becomes very large and negative. A negative number cubed stays negative and grows in magnitude.

So

$y\to-\infty.$

(iii) $y=\frac{1}{x}$ as $x\to-\infty$

As $x$ becomes a very large negative number, the reciprocal gets closer to zero.

So

$y\to 0.$

If you want to be extra precise, it approaches $0$ from below.

💡 Pitfall guide

Do not confuse the sign inside $(2-x)^3$. For large $x$, the expression is not positive just because it is cubed. The inner term $2-x$ is negative.

Also, for $\frac{1}{x}$, do not say the output becomes very large in magnitude. It actually shrinks toward zero.

🔄 Real-world variant

If part (ii) were $y=(2-x)^4$ instead, then the output would go to $+\infty$ because an even power removes the negative sign. If part (iii) were $y=\frac{1}{x^2}$ as $x\to-\infty$, the result would still be $0$, but from the positive side.

🔍 Related terms

end behavior, power function, reciprocal graph