Question

Select the correct options for the graphs of the functions $f\left(x\right)=a^x;a>1\&$$g\left(x\right)=b^x;0<b<1.$

• A. $f\left(x\right)$ is a constant function
• B. $g\left(x\right)$ is a strictly decreasing function
• C. $\frac{1}{g\left(x\right)}$ is a strictly increasing function
• D. $f\left(x\right)$ is a strictly increasing function

Answer 1

Answer: (B), (C), (D)

$f\left(x\right)$ is a strictly increasing function

We have, $f\left(x\right)=a^x;a>1$
We know that as we increses the value of $x$ the value of $f\left(x\right)$ increases, so the graph of $f\left(x\right)=a^x;a>1$ goes up. We can also see that from the below graph.

Similarly, we have $g\left(x\right)=b^x;0<b<1$
We know that as we increses the value of $x$ the value of $g\left(x\right)$ decreases, so the graph of $g\left(x\right)=b^x;0<b<1$ looks like falling down. We can also see that from the below graph.

Now, talking about $\frac{1}{g\left(x\right)}=\frac{1}{b^x}$
Since, $0<b<1\Rightarrow \infty >\frac{1}{b}>1$
$\Rightarrow p\left(x\right)=\frac{1}{g\left(x\right)}=\frac{1}{b^x}=(\frac{1}{b}{)}^x=k^x;k>1$
Thus, this function will be strictly increasing function as well.