Question

Let f : R \setminus \left\\{ -\frac{1}{2} \right\\} \to R and g : R \setminus \left\\{ -\frac{5}{2} \right\\} \to R be defined as $f(x) = \frac{2x + 3}{2x + 1}$ and $g(x) = \frac{|x| + 1}{2x + 5}$. Then the domain of the function $f(g(x))$ is:

• A. R \setminus \left\\{ -\frac{5}{2} \right\\}
• B. $R$
• C. R \setminus \left\\{ -\frac{7}{4} \right\\}
• D. R \setminus \left\\{ -\frac{5}{2}, -\frac{7}{4} \right\\}

Answer 1

Answer: (A)

R \setminus \left\\{ -\frac{5}{2} \right\\}

Answer 2

For the function composition $f \circ g(x)$ to be defined, $g(x)$ must be in the domain of $f$. Given:
$f(g(x)) = \frac{2g(x) + 3}{2g(x) + 1}.$

The domain of $f$ excludes $x = -\frac{3}{2}$. Therefore, $2g(x) + 1 \neq 0$. Solving:
$2\left(\frac{|x| + 1}{2x + 5}\right) + 1 = 0.$

Simplifying:
$\frac{2(|x| + 1)}{2x + 5} + 1 = 0 \implies |x| + 1 = -\frac{2x + 5}{2}.$

This yields no valid solutions in $\mathbb{R}$. Therefore, the domain is:
\mathbb{R} \setminus \left\\{\frac{-5}{2}\right\\}.

Thus, the correct answer is: R \setminus \left\\{ -\frac{5}{2} \right\\}