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Write an equation that defines the exponential function with base a>0 ?

Here is the equation of exponential equation with base a>0.

y=2^{x-1}

Domain of this Function

(0,+\infty)

Range of this Function

Real Numbers

(c) If the a is not equal to 1 then its Domain and Range will also remains same.

Sketch the general shape of graph of the exponential function for each of the following cases?

Define the bracket function?

Bracket function is also called as Greatest Integer Function or Step Function. It is defined as the nearest and greatest integer nearer to the given real number.

Draw the graph of [x]+2x-3.

Domain and Range of [x]+2x-3 are

• Domain = Real Numbers
• Range = All Real Numbers

Define the Absolute Value function?

The function that contains algebraic expression x in absolute value symbol |x| .

Graph of signum function is

• Domain = Real Numbers except 0

• Range = {-1,1}

Domain and Range of following functions:

(i)  \frac{1}{1+e^x}

• Domain = Real Numbers

• Range = Real Numbers

(ii)  \sin\left(e^{-t}\right)

• Domain = Real Numbers

• Range = [-1, 1]

(iii)  \frac{1}{\left|x-4\right|}

• Domain = Real Numbers except 4

• Range = (0,+∞)

(iii)  \frac{x-2}{x+3}

• Domain = Real Numbers except -3

• Range = Real Numbers