Animated Solution for Mathematics - Functions: Let function $f:R\to R$ be defined by $f(x)=2x+\sin x$ for $x\in R$, then $f$ is

Imagine you are standing on a path that is steadily rising, but with every step, you feel a gentle, rhythmic sway. This is the essence of the function $f(x)=2x+\sin x$.

It is a beautiful marriage of a linear backbone, $2x$, and a periodic oscillation, $\sin x$. To understand if this function is one-to-one and onto, we must look beyond the surface and into its rate of change.

For a function to be one-to-one (injective), it must never "look back." If you draw a horizontal line anywhere on its graph, it should intersect the curve at most once.

Mathematically, we ensure this by checking if the function is strictly monotonic. We do this by finding the derivative, $f^{\prime }(x)$.

Differentiating our function, we get:

Now, consider the behavior of $\cos x$. We know that for any real number $x$, the value of $\cos x$ is trapped between $-1$ and $1$.

By adding $2$ to this inequality, we find the range of our derivative:

Because $f^{\prime }(x)\ge 1$, the derivative is always strictly positive. This means the slope of the function is always at least $1$.

It never becomes zero, and it certainly never becomes negative. The function is always climbing, never pausing, and never descending. Therefore, it is strictly increasing, which confirms that $f(x)$ is indeed one-to-one.

Now, let's tackle the "onto" (surjective) property. A function is onto if its range covers the entire codomain, which in this case is the set of all real numbers, $\mathbb{R}$.

We need to see if the function can reach every possible $y$-value. Let's look at the limits at the boundaries of our domain:

As $x$ approaches positive infinity, the linear term $2x$ dominates, driving the function to infinity. As $x$ approaches negative infinity, the function is dragged down to negative infinity.

Because $f(x)$ is a continuous function, the Intermediate Value Theorem guarantees that it must pass through every single value between $-\infty$ and $\infty$. Thus, the range is $(-\infty ,\infty )$, which is exactly the codomain $\mathbb{R}$.

The Perfect Bijective Pair

We have proven that our function is strictly increasing (making it one-to-one) and that it spans the entire set of real numbers (making it onto).

A function that is both one-to-one and onto is called a bijection.

You have successfully navigated the geometry and calculus of this function, revealing its elegant, bijective nature. Keep this analytical mindset, and you will find that even the most complex functions have a story to tell.