Question

How do you graph $ f\left( x \right)={{2}^{x-1}}-3 $ and state the domain and range?

Answer 1

Hint: We first describe the use of a function. From the given function we find the restrictions on $ x $ which in turn helps in finding the domain. The whole function is then put under the domain to find the range.

Complete step-by-step answer:
The basic requirement for a function to have input and evaluate its output. The input is called the domain of the function and the outcome is called the range.
We generally define a function as $ f\left( x \right) $ . We can also express it as
 $ f:x\to f\left( x \right) $ .
Here for the given function, we have $ f\left( x \right)={{2}^{x-1}}-3 $ . The power value of 2 can not have any restrictions. Therefore, we can use any real value for the $ x $ .
The domain for the function $ f\left( x \right)={{2}^{x-1}}-3 $ is $ \mathbb{R} $ .
Now we have to find the range for the function.
We know $ \forall x\in \mathbb{R} $ , the value of $ {{2}^{x-1}}>0 $ . We subtract the value 3 from the inequation of $ {{2}^{x-1}}>0 $ .
We get $ \forall x\in \mathbb{R} $ , the value of $ {{2}^{x-1}}-3>-3 $ .
The equality is not possible as there exists no $ x\in \mathbb{R} $ for which $ {{2}^{x-1}}=0 $ .
Therefore, $ \forall x\in \mathbb{R} $ , the value of $ f\left( x \right)={{2}^{x-1}}-3>-3 $ . The range of the function is $ \left( -3,\infty \right) $ .

Note: We have to be careful about the equality of the function’s equality for the boundary points. If the boundary points are not in the domain then the function would not be defined at that point.