Question

Which one of the following is not correct for the feature of an exponential function given by $ f\left( x \right)={{b}^{x}} $ , where $ b>1 $ ?
A. For very large negative values of $ x $ , the function is very close to 0.
B. The domain of the function is R, the set of real numbers.
C. The point $ \left( 1,0 \right) $ is always on the graph of the function.
D. the range of the function is the set of all positive real numbers.

Answer 1

Hint: We take the function and find its range and domain. We explain the given options and check if they are applicable for the function. We assume negative form for the very large negative values of $ x $ and find its limit value.

Complete step-by-step answer:
We check every given option for the exponential function
$ f\left( x \right)={{b}^{x}} $ , where $ b>1 $ .
For very large negative values of $ x $ , we assume $ x=-m,m\to \infty $ . So,
$ {{b}^{x}}={{b}^{-m}} $ .
So, $ {{b}^{x}}={{b}^{-m}}=\dfrac{1}{{{b}^{m}}} $ . As $ m\to \infty ,{{b}^{m}}\to \infty $ . Therefore, $ {{b}^{x}}=\dfrac{1}{{{b}^{m}}}\to 0 $ .

The function $ f\left( x \right)={{b}^{x}} $ is valid for every domain and that’s why the domain is R.

We now check if $ \left( 1,0 \right) $ satisfies the $ f\left( x \right)={{b}^{x}},b>1 $ . If we put the values $ {{b}^{1}}=b=0 $ .
But it is given that $ b>1 $ . Therefore, the point $ \left( 1,0 \right) $ is not on the graph of the function.

As $ b>1 $ , the only solution for \[{{b}^{x}}=0\] is $ b=0 $ which is not possible. Therefore, the range doesn’t consist of 0.
The last two options C and D are not correct for the feature of an exponential function given by $ f\left( x \right)={{b}^{x}} $ , where $ b>1 $ .
So, the correct answer is “Option C and D”.

Note: We need to remember that we can also change the function of logarithm form by taking $ \log y=\log \left( {{b}^{x}} \right)=x\log b $ . The function becomes dependent on the value of $ x $ directly. Exponential function is with range \[\left( 0,\infty \right)\].