Question

Above shown is the graph of the function $f\left(x\right)$. If there is a function $g\left(x\right)$ such that $g\left(x\right)=\frac{1}{x}$, then find the true statement for $(g\circ f)\left(x\right)$.

• A. $(g\circ f)\left(x\right)$ is defined for all real numbers.
• B. $(g\circ f)\left(x\right)$ is undefined for three real values of x.
• C. $(g\circ f)\left(x\right)$ is undefined for four real values of x.
• D. $(g\circ f)\left(x\right)$ is undefined for all real numbers.

Answer 1

The correct option is C $(g\circ f)\left(x\right)$ is undefined for four real values of x.

$g\left(x\right)=\frac{1}{x}$

So, $g\left(x\right)$ is undefined for $x=0$

$\left(gof\right)\left(x\right)=g\left(f\right(x\left)\right)=\frac{1}{f\left(x\right)}$

So, $\left(gof\right)\left(x\right)$ is undefined for $f\left(x\right)=0$

From the graph we can see that $f\left(x\right)=0$ has four solutions.

So, the function $\left(gof\right)\left(x\right)$ is undefined for four values of $x$

So, the answer is option (C)