Animated Solution for Mathematics - Functions: Let $f,g:\mathbb{R}\to \mathbb{R}$ be defined as : $f(x)=|x-1|$ and $g(x)=\{\begin{array}{ll}{e}^x,& x\ge 0\\ x+1,& x\le 0\end{array}$. Then the function $f(g(x))$ is

Welcome, fellow traveler, to the beautiful world of composite functions! Today, we are going to dissect a problem that perfectly illustrates how functions interact.

We are given $f(x)=|x-1|$ and a piecewise function $g(x)$. Our goal is to understand the nature of $h(x)=f(g(x))$.

Think of $h(x)$ as a two-stage machine: first, the input $x$ passes through the $g$ machine, and then the output of that machine is fed into the $f$ machine.

Our master equation is $h(x)=|g(x)-1|$. Because $g(x)$ is defined differently for positive and negative inputs, our composite function $h(x)$ must also be defined piecewise.

Let us look at the case where $x\ge 0$. Here, $g(x)=e^x$. Substituting this into our master equation, we get:

Now, consider the behavior of $e^x$ for $x\ge 0$. Since $e^x\ge e^0=1$, the expression $e^x-1$ is always non-negative.

Thus, the modulus simply vanishes, leaving us with $h(x)=e^x-1$. This is the right branch of our function, starting from the origin and climbing rapidly toward infinity.

Now, let us shift our focus to the region where $x<0$. In this domain, $g(x)=x+1$.

Substituting this into $h(x)=|g(x)-1|$, we get:

The $+1$ and $-1$ cancel out, leaving us with $h(x)=|x|$. Since we are in the domain $x<0$, the definition of the modulus tells us that $|x|=-x$.

So, for the left side, our function is simply the line $h(x)=-x$. This is a straight line descending from infinity to the origin.

We have now constructed our function:

If you visualize this, you see a V-shaped curve that touches the origin.

Now, let us test for injectivity (one-one). If we draw a horizontal line at $y=2$, it intersects the graph at two points: one on the left branch and one on the right. Because a horizontal line hits the graph twice, the function is not one-one.

Finally, let us test for surjectivity (onto). The graph starts at $y=0$ and goes up to infinity. It never touches the negative $y$-axis.

Since the Range is $