Question

The interval for which the function $f\left(x\right)=2-a^x>0,\forall a\in (0,1)$ is

• A. $(-\infty ,1)$
• B. $(-1,\infty )$
• C. $(1,\infty )$
• D. $(-\infty ,\infty )$

Answer 1

The correct option is B $(-1,\infty )$

We are given the function $f\left(x\right)=2-a^x,\forall a\in (0,1)$
To find the interval where $f\left(x\right)>0$, let us draw the graph of $f\left(x\right)$.
First we draw the graph of $a^x\forall a\in (0,1)$
Now we invert this graph on the x-axis to get the graph of $-a^x\forall a\in (0,1)$
Now we shift the graph upwards by $2$ units to finally get the graph of $f\left(x\right)=2-a^x,\forall a\in (0,1)$
From the graph we observe that $f\left(x\right)>0\forall x>-1$, hence the correct interval is $(-1,\infty )$