Animated Solution for Mathematics - Functions: Let $f(x)={\log }_{e}x$ and $g(x)=\frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}$. Then the domain of $f(g(x))$ is

We are tasked with finding the domain of the composite function $f(g(x))$, where $f(x)=\ln (x)$ and $g(x)=\frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}$.

To determine the domain, we must satisfy the condition that the inner function $g(x)$ is defined and that its output is a valid input for the outer function $f(x)$.

The Logarithmic Constraint

The natural logarithm function $f(x)=\ln (x)$ is defined only for strictly positive arguments. Therefore, for the composite function $f(g(x))$ to exist, we must satisfy the inequality:

This condition serves as the primary constraint for our domain.

The Safety Check

Let the denominator of $g(x)$ be $D(x)=2x^2-2x+1$. To check for potential zeros or negative values, we calculate the discriminant $\Delta =b^2-4ac$:

Since $\Delta <0$ and the leading coefficient $a=2$ is positive, the quadratic $D(x)$ is always positive for all $x\in \mathbb{R}$. Thus, the denominator never vanishes, and $g(x)$ is defined for all real numbers.

The Algebraic Magic

Now we examine the numerator $N(x)=x^4-2x^3+3x^2-2x+2$. We can decompose this expression to reveal its behavior:

This simplifies to:

Alternatively, by grouping terms differently, we observe:

Since $(x^2-x+1{)}^2\ge 0$ for all real $x$, it follows that $N(x)\ge 1$. The numerator is strictly positive for all real values of $x$.

A Domain Without Borders

We have established that for all $x\in \mathbb{R}$, the numerator $N(x)$ is positive and the denominator $D(x)$ is positive. Consequently, the ratio $g(x)=\frac{N(x)}{D(x)}$ is strictly positive for all $x\in \mathbb{R}$.

Because $g(x)>0$ is satisfied for every real number, the logarithmic constraint is always met. There are no values of $x$ that must be excluded from the domain.