Question

If $f : B \to A$ is defined by $f \left(x\right) = \frac{3x+4}{5x-7}$ and $g : A \to B$ is defined by $g \left(x\right) = \frac{7x+4}{5x-3}$, where A = R - \left\\{\frac{3}{5}\right\\} and B = R - \left\\{\frac{7}{5}\right\\} and $I_{A}$ is an identity function on A and $I_{B}$ is identity function on B, then

• A. $fog = I_A$ and $gof = I_A$
• B. $fog = I_A$ and $gof = I_B$
• C. $fog = I_B$ and $gof = I_B$
• D. $fog = I_B$ and $gof = I_A$

Answer 1

Answer: (B)

$fog = I_A$ and $gof = I_B$

Answer 2

We have, $gof \left(x\right) = g\left(\frac{3x+4}{5x-7}\right) = \frac{7\left(\frac{\left(3x+4\right)}{\left(5x-7\right)}\right)+4}{5\left(\frac{\left(3x+4\right)}{\left(5x-7\right)}\right)-3}$ $= \frac{21x + 28 + 20x - 28}{15x + 20 - 15x + 21} = \frac{41x}{41} = x$ Similarly, $fog \left(x\right) = f\left(\frac{7x+4}{5x-3}\right)$ $= \frac{3\left(\frac{\left(7x+4\right)}{\left(5x-3\right)}\right)+4}{5\left(\frac{\left(7x+4\right)}{\left(5x-3\right)}\right)-7}$ $= \frac{21x + 12 + 20x - 12}{35x + 20 - 35x + 21} = \frac{41x}{41} = x$ Thus, $gof \left(x\right) = x, \forall x \in B$ and $fog \left(x\right) = x, \forall x \in A$, which implies that $gof = I_{B}$ and $fog = I_{A}.$