Question

Which one of the following is not correct for the features of exponential given by $f\left(x\right)=b^x$ where $b>1$?

• A. The range of the function is $R$, the set of real numbers.
• B. The range of the function is the set of all positive real numbers.
• C. For very large negative values of $x$, the functions is very close to $0$.
• D. The point $(0,1)$ is always on the graph of the function.

Answer 1

The correct option is A The range of the function is $R$, the set of real numbers.

Consider the function $f\left(x\right)=b^x$. The domain of the function is set of all real numbers. Therefore, $x\in R$.
Now $y=b^x$,

$\log y=x\log b$. Here, $b>1$ and the input of the log cannot be $0$. Hence $y>0$. Therefore, the range of $f\left(x\right)>0$.
Now for $x\to \infty ,$ $f\left(x\right)\to \infty$. Also, for $x\to -\infty$, $f\left(x\right)\to 0$. Hence for large positive real numbers, the function tends to infinity.

Similarly for negative numbers with large magnitudes, the function tends to be $0$.
Also, $f\left(1\right)=b^1=b$ and $f\left(0\right)=b^0=1$. Therefore, $(1,b)$ and $(0,1)$ lies on the graph of the function.