Let f(x) = sin^{-1} x and g(x) = frac{x^2 - x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The domain of $(f \circ g)(x) = \sin^{-1}(g(x))$ requires $|g(x)| \leq 1$.First,we find $g(2) = \lim_{x 	o 2} \frac{(x+1)(x-2)}{(2x+3)(x-2)} = \l

Vedclass · Accepted Answer

$(-\infty, -2] \cup [-\frac{4}{3}, \infty)$

Vedclass · Answer

(C) The domain of $(f \circ g)(x) = \sin^{-1}(g(x))$ requires $|g(x)| \leq 1$.First,we find $g(2) = \lim_{x 	o 2} \frac{(x+1)(x-2)}{(2x+3)(x-2)} = \lim_{x 	o 2} \frac{x+1}{2x+3} = \frac{3}{7}$.Thus,we solve $|\frac{x+1}{2x+3}| \leq 1$ for $x 
eq 2$.This implies $-1 \leq \frac{x+1}{2x+3} \leq 1$.Case $1$: $\frac{x+1}{2x+3} \leq 1 \Rightarrow \frac{x+1 - (2x+3)}{2x+3} \leq 0 \Rightarrow \frac{-x-2}{2x+3} \leq 0 \Rightarrow \frac{x+2}{2x+3} \geq 0$.The solution is $x \in (-\infty, -2] \cup (-\frac{3}{2}, \infty)$.Case $2$: $\frac{x+1}{2x+3} \geq -1 \Rightarrow \frac{x+1 + 2x+3}{2x+3} \geq 0 \Rightarrow \frac{3x+4}{2x+3} \geq 0$.The solution is $x \in (-\infty, -\frac{4}{3}] \cup (-\frac{3}{2}, \infty)$.Taking the intersection of both cases and excluding $x=2$ (since $g(2)$ is defined as the limit),we get $x \in (-\infty, -2] \cup [-\frac{4}{3}, \infty)$.

Let $f(x) = \sin^{-1} x$ and $g(x) = \frac{x^2 - x - 2}{2x^2 - x - 6}$. If $g(2) = \lim_{x \to 2} g(x)$,then the domain of the function $f \circ g$ is .... .

Similar Questions

If $f(x) = \frac{4x + 3}{6x - 4}$, $x \neq \frac{2}{3}$ and $(f \circ f)(x) = g(x)$ where $g : R - \{\frac{2}{3}\} \to R - \{\frac{2}{3}\}$, then $(g \circ g \circ g \circ g \circ g)(3) = $

If $f(x) = ax + b$ and $g(x) = cx + d$,then $f(g(x)) = g(f(x))$ is equivalent to

If $g(x) = x^2 + x - 2$ and $\frac{1}{2} (g \circ f)(x) = 2x^2 - 5x + 2$,then $f(x)$ is equal to:

If $f(x)=3x+6$,$g(x)=4x+k$ and $f \circ g(x)=g \circ f(x)$,then $k =$

Let $f(x)$ and $g(x)$ be two real polynomials of degree $2$ and $1$ respectively. If $f(g(x)) = 8x^2 - 2x$ and $g(f(x)) = 4x^2 + 6x + 1$,then the value of $f(2) + g(2)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module